An Algorithmic Approach to Algebraic and Dynamical Cancellations associated to a Spectral Sequence
Maria Alice Bertolim, Dahisy V. S. Lima, Margarida P. Mello, Ketty A. de Rezende, Mariana R. da Silveira
AAn Algorithmic Approach to Algebraic and DynamicalCancellations associated to a Spectral Sequence
M.A. Bertolim D.V.S. Lima M.P. Mello K.A. de Rezende M. R. da Silveira Abstract
In this article we study algorithms that arise in both topological and dynamical settings, namely, theSpectral Sequence Sweeping Algorithm (SSSA) and the Row Cancellation Algorithm (RCA) for a filteredMorse chain complex on a manifold M n . Both algorithms have as input a connection matrix and the resultsobtained in this article make it possible to establish a correspondence between the algebraic cancellations inSSSA and the dynamical cancellations in RCA. Key words: connection matrix, Morse complex, spectral sequence, integer programming.
Introduction
Algorithms for computing spectral sequences are well known in the literature and have been implemented byseveral authors [11]. For example, in the field of Computational Topology, see [6], there has been much interestin algorithms which solve problems anywhere from the reconstruction of surfaces in computer graphics to thetreatment of noise in data input. For example, to gather information about a surface using a computer, it isnecessary to make use of a combinatorial representation. This information can be provided by making use ofsimplicial complexes. Data structures are then constructed with the purpose of storing cell complex information.To retrieve topological connectivity information from this data, homology and spectral sequences are used.We approach spectral sequences from a dynamical systems point of view. Our primary interest resides inrelating qualitative aspects of dynamical systems which can be coded algebraically in a chain complex description,to topological aspects of the phase space and the asymptotic behavior of stable and unstable manifolds. Thestudy of the latter, in a one-parameter family of flows, using a homotopical tool such as the Conley index,permits a deeper understanding of bifurcation behavior in a continuation, which is reflected, for example, inbirth and death of critical points. There are many techniques that may be used to achieve such an endeavor.The underlying approach used here has been to bridge the algebraic-topological and dynamical realms by use ofspectral sequences.In order to understand this more fully, let f : M → R be a Morse function on a closed Riemannian n -manifold M and consider the Morse chain complex ( C, ∆) where the Morse chain group C = { C k ( f ) } is defined as thefree abelian group generated by the critical points of f and graded by their Morse indices, i.e., C k ( f ) = (cid:77) x ∈ Crit k ( f ) Z [ x ]where Crit k ( f ) denotes the set of index k critical points of f . In this case, the differential ∆ : C → C is acollection of homomorphisms ∆ k : C k ( f ) → C k − ( f ) which are defined on a generator x ∈ Crit k ( f ) by∆ k ( x ) = (cid:88) y ∈ Crit k − ( f ) n ( x, y )[ y ] , Supported by FAPESP under grant 2010/08579-0. Partially supported by CNPq under grant 302592/2010-5 and by FAPESP under grant 2012/18780-0. Partially supported by FAPESP under grant 2012/18780-0. a r X i v : . [ m a t h . D S ] A ug here n ( x, y ) denotes the intersection number of x and y .A spectral sequence E = ( E r , d r ) r ≥ is a sequence, such that E r is a bigraded R -module over a principal idealdomain R , i.e., an indexed collection of R -modules E rp,q for all pair of integers p and q ; and d r is a differential of bidegree ( − r, r − d r : E rp,q → E rp − r,q + r − for all p and q such that d r ◦ d r = 0. Moreover, for all r ≥ H ( E r ) ≈ E r +1 where H p,q ( E r ) = Ker d r : E rp,q → E rp − r,q + r − Im d r : E rp + r,q − r +1 → E rp,q is the homology module. For more details see [14].One way to visualize a spectral sequence is as a book consisting of pages such that the r -th page correspondsto the bigraded module E r . On each page there are homomorphisms between the modules forming sequences ofchain complexes. Moreover, the homology module of the r -th page is precisely the ( r + 1)-th page. See Figure 1. E , E , E , E , E , E , E , E , E , E , E , E , E , E , E , E , E E , E , E , E , E , E , E , E , E , E , E , E , E , E , E , E , E E , E , E , E , E , E , E , E , E , E , E , E , E , E , E , E , E d - bidegree (0 , − d - bidegree ( − , d - bidegree ( − , d d d E , E , E , E , E , E , E , E , E , E , E , E , E , E , E , E , E d d - bidegree ( − , · · ·· · ·· · ·· · ·· · ·· · ·· · ·· · · · · ·· · ·· · ·· · ·· · ·· · ·· · ·· · · ... ... ... ... ... ... ... ...... ... ... ...... ... ... ... Figure 1: Spectral sequence pages.In this paper, one considers a specific filtration in a Morse complex ( C, ∆) defined by f . Given a finest Morsedecomposition { M ( p ) | p ∈ P = { , . . . , m } , m = Crit ( f ) } such that there are distinct critical values c p with f − ( c p ) ⊃ M ( p ), we can define a filtration on M by { F p − } mp =1 = { f − ( −∞ , c p + (cid:15) ) } mp =1 . Let F = F p C the induced filtration in ( C, ∆). Since for each p ∈ P there is only one singularity in F p \ F p − thefiltration F is called a finest filtration.With this in mind, we consider a filtered Morse chain complex ( C, ∆) and calculate its spectral sequence.In this case the differential ∆ is a connection matrix as proved in [12]. Nonzero entries of this matrix implythe existence of connecting orbits, hence connection matrices provide information concerning global dynamics.In this article, with a view towards application, we develop algorithms that model the spectral sequence of2 filtered Morse chain complex in the hope of coming across a relationship between algebraic cancellation ofmodules and dynamical cancellation of critical points. These algorithms are intimately related to the SpectralSequence Sweeping Algorithm (SSSA) developed in [3, 4, 5, 8], and are introduced with the intention of allowingone to gain a deeper understanding of the role of the differentials.It was proved in [4] that the Spectral Sequence Sweeping Algorithm applied to a connection matrix ∆generates a sequence of connection matrices ∆ r which keeps track of the differentials d r that cause the algebraiccancellations of modules E r of the spectral sequence, for r ≥
0. Of course, this immediately raises the questionof understanding up to what point algebraic cancellations in ( E r , d r ) determine dynamical cancellations in aparameterized family of flows.In [3], a parameterized family of Morse flows on surfaces that undergo bifurcations was associated to thissequence of connection matrices ∆ r . In other words, for the 2-dimensional case, the algebraic cancellations thatoccur as one “turns the pages” E r of the spectral sequence correspond to the dynamical cancellations that mayoccur in this parameterized family of flows. More specifically, the differentials of the spectral sequence that causealgebraic cancellations as r increases are in 1-to-1 correspondence with the primary pivots of ∆ r determinedby the Spectral Sequence Sweeping Algorithm. It is interesting to note that the connection matrices in the2-dimensional setting are totally unimodular (TU).This instigated our inquiry into the higher dimensional case under the hypothesis that the connection matrix∆ be totally unimodular. In this case, one can assert that for a filtered Morse chain complex ( C, ∆) on a closedsimple connected manifold M of dimension m >
5, with ∆ a TU connection matrix, the algebraic cancellationsof the modules of the spectral sequences correspond to dynamical cancellations of consecutive critical pointsdetermined by the Row Cancellation Algorithm (RCA).This result was first obtained in [3] with the proviso that these algorithms were compatible. Among otheringredients of this proof, two propositions were needed which are proved herein. The first being that the primarypivots were ±
1, and the second that the primary pivots in both algorithms coincide in position and value, seeCorollaries 30 and 31. In attempting to prove this in the 2-dimensional case, we actually obtained proofs indimension n , which are described in Theorems 18 and 28 and enabled us to prove Theorem 1. Hence, theseconstitute the main results in this article, the first two are more algebraic in nature whereas the last has a moredynamical flavor.The main dynamical result in this article is the following. Theorem 1
Let ( C, ∆) be a Morse chain complex associated to a Morse-Smale function f on a closed simplyconnected manifold M of dimension m > . Suppose that ∆ is totally unimodular. Let ( E r , d r ) r ≥ be theassociated spectral sequence for the finest filtration F = { F p C } defined by f . The algebraic cancellations of themodules E r of the spectral sequence determined by the SSSA are in one-to-one correspondence with dynamicalcancellations that occur in Morse flows on M associated to ( C, ∆) determined by the RCA. Corollary 2
Let ( C, ∆) be a Morse chain complex associated to a Morse-Smale function f on a closed simplyconnected manifold M of dimension m > . Suppose that ∆ is a TU connection matrix then admits a perfectMorse function. Several questions are in order here. Of course, the question of considering this result without the TUhypothesis for arbitrary connection matrices in dimension n remains open. We mean by Morse flows those that arise from a gradient of a Morse function and satisfy the transversality condition. h h h h h h h h h h F F F F F F F F F F h h h h h h F F F F F F F F F F h h Cancelling ( h , h )( h , h ) Cancelling ( h , h )( h , h ) Figure 2: Continuation via cancellation of critical points. E : E : E : [ h ] [ h ] [ h ] [ h ] [ h ] [ h ] E E E E E E [ h ] E [ h ] E [ h ] E [ h ] E E E E E E E d d E : E E E E E E d d E : E E Figure 3: Spectral sequence of a filtered Morse chain complex.4
Sweeping algorithms
In this section we present several sweepings algorithm for connection matrices. We adopt the following notation,namely, that the columns and rows of the order m connection matrix ∆ are partitioned into the subsets J , . . . , J b , columns and rows in J k are associated with elementary chains of index k . A connection matrix is groupedwhenever J i is a set of consecutive integers, for all i , with the entries in J k − × J k located above the diagonal,and ungrouped otherwise.The motivation behind working with ungrouped connection matrices lies in the fact that the data of asimplicial complex is generally presented in a disorderly fashion, i.e., k − simplices unordered with respect to agiven filtration. Hence, columns representing index k -critical points appear with mixed indices from left to right.See Figure 4.Notation regarding matrices is defined in Table 1. A i . i -th row of matrix AA . j j -th column of matrix AA I . submatrix of A with entries a ij such that i ∈ IA IJ submatrix of A with entries a ij such that i ∈ I and j ∈ J ,where I (resp., J ) is a nonempty subset of the set ofrow indices (resp., column indices) A (cid:96) (cid:96) -th matrix in a sequence, non negative superscripts do not denote exponents( A (cid:96) ) − the inverse of matrix A (cid:96) Table 1: Notation adopted for (sub)matrices.The connection matrix ∆ is an upper triangular nilpotent matrix with zero entries except, possibly, for theblocks ∆ J k − ,J k , for k = 1 , . . . , b . In [5], the connection matrices considered were grouped, i.e., all blocks weresituated strictly above the main diagonal, and thus each subset in the partition contained consecutive indices. Forexample, Figure 4 shows two possible configurations for matrices with indices partitioned into four subsets. Bothmatrices have index subsets of same cardinality (3, 5, 2 and 2), but the one on the left has consecutive indiceswithin each subset, whereas the one on the right does not. Positions in the first block are indicated with verticallines, in the second with horizontal lines and in the third with diagonal lines. Shaded areas indicate positionsthat may have nonzero entries, the allowable sparsity pattern of the connection matrix, namely positions in theset ∪ b − k =0 J k × J k +1 that lie strictly above the diagonal. Notice that, since the matrix must be upper triangular,the scattering reduces the number of positions eligible for nonzero entries. Thus, in the example on the left ofFigure 4, the connection matrix may have a total of 29 nonzero entries, but the number of nonzero entries of thematrix on the right is at most 17. We call a connection matrix with blocks above the main diagonal a groupedconnection matrix , to distinguish it from the (general) connection matrix exemplified on the right of Figure 4.Of course, the grouped connection matrix is a special type of connection matrix. The set of consecutive integers { i, i + 1 , . . . , i + k } is denoted i..i + k .The sweeping algorithm over Z was introduced in [4] and further explored in [5]. It was stated in terms ofgrouped connection matrices, and we rewrite it below, making the necessary slight notation changes to encom- When there is no danger of ambiguity, the comma between row and column indices is omitted. (cid:122) (cid:125)(cid:124) (cid:123) J (cid:122) (cid:125)(cid:124) (cid:123) J (cid:122)(cid:125)(cid:124)(cid:123) J (cid:122)(cid:125)(cid:124)(cid:123) (cid:122)(cid:125)(cid:124)(cid:123) J (cid:122)(cid:125)(cid:124)(cid:123) J (cid:122)(cid:125)(cid:124)(cid:123) J (cid:122)(cid:125)(cid:124)(cid:123) J J = 1 .. J = 4 .. J = 9 .. J = 11 .. J = { , , } , J = { , , , , } , J = { , } , J = { , } Figure 4: Two possible configurations of matrices with column/row sets partitioned into 4 subsets.pass the general case.
Sweeping Algorithm over Z Input: nilpotent m × m upper triangular matrix ∆ with column/row partition J , J , . . . , J b . Initialization Step: r = 0∆ r = ∆ P r = I ( m × m identity matrix) Iterative Step: (Repeated until all diagonals parallel and to the right of the main diagonal have been swept)
Matrix ∆ update r ← r + 1∆ r = ( P r − ) − ∆ P r − Markup
Sweep entries of ∆ r in the r -th diagonal: If ∆ rj − r,j (cid:54) = 0 and ∆ r . ,j does not contain a primary pivot Then If ∆ rj − r . contains a primary pivot Then temporarily mark ∆ rj − r,j as a change-of-basis pivot Else permanently mark ∆ rj − r,j as a primary pivot6 Matrix P r computation P r ← P r − For each change-of-basis pivot ∆ rj − r,j , update the j -th column of P r as followsLet k be the index of the chain associated with column j of ∆.Let I = J k − ∩ { j − r, . . . , m } , J = J k ∩ { , . . . , j } , c = | J | Let x ∗ ∈ Z c be an optimal solution tomin x c subject to ∆ IJ x = 0 x c ≥ x ∈ Z c P rJj ← x ∗ Final Step:
Matrix ∆ update r ← r + 1∆ r = ( P r − ) − ∆ P r − Output: (∆ , . . . , ∆ m ) and ( P , . . . , P m − )The connection matrix can be thought of as the matrix of a linear operator with respect to the basis h =( h i , . . . , h mi m ), where i j denotes the index of the chain associated with column j . The sweeping algorithm producesa sequence of similar matrices ∆ (= ∆), ∆ , . . . , ∆ m . The basis associated with ∆ r is σ r = ( σ ,ri , . . . , σ m,ri m ).Initially, we have σ = h . At iteration r of the sweeping algorithm, the entries on the r -th diagonal of ∆ r areswept. Zero entries are left unmarked. A nonzero entry at position ( j − r, j ) is left unmarked if there is a primarypivot below it, is marked temporarily as a change-of-basis pivot if there is a primary pivot to its left, and isotherwise permanently marked as a primary pivot. If ∆ rj − r,j is a change-of-basis entry, then the element σ r +1 j inthe new basis σ r +1 is an integer linear combination of the elements of h with the same chain index, say k , andassociated with columns to the left of, and including, column j , constructed so that the following two conditionsare satisfied: (1) the entries on and below position ( j − r, j ) in ∆ r +1 are zero, and (2) the coefficient of h jk inthis linear combination, called the leading coefficient , is the smallest positive integer that allows (1) to happen.The correctness of the sweeping algorithm over Z rests on Proposition 4, an extension of Proposition 8 of [5]that encompasses both general and grouped connection matrices. Before proceeding, we introduce a technicallemma that facilitates the extension of the results previously developed for grouped connection matrices. Lemma 3
Let N be an upper triangular m × m matrix with nonzero diagonal entries and column/row partition J , . . . , J b , such that n ij = 0 if ( i, j ) / ∈ J k × J k , for k = 0 , . . . , b . Then N is invertible, its inverse is uppertriangular and also has zero entries outside the set ∪ bk =0 J k × J k . Furthermore, ( N J k J k ) − = ( N − ) J k J k , for k = 0 , . . . , b. Additionally, if j ∈ J k , for some k ∈ { , . . . , b } , I = J k ∩ { j, . . . , m } and J = J k ∩ { , . . . , j } , I and J denotetheir respective complements with respect to { , . . . , m } , then ( N − ) II = 0 and ( N − ) JJ = 0 . Temporary marks are erased at the end of the iterative step. roof . If N is upper triangular, its determinant is the product of the diagonal entries. Since these entries arenonzero by hypothesis, N is nonsingular and its inverse is upper triangular. Let the column/row partition begiven by J = { i , . . . , i | J | } , where i < · · · < i | J | J = { i | J | +1 , . . . , i | J | + | J | } , where i | J | +1 < · · · < i | J | + | J | , ... J b = { i | J | + ··· + | J b − | +1 , . . . , i | J | + ··· + | J b | } , where i | J | + ··· + | J b − | +1 < · · · < i | J | + ··· + | J b | , and let p : { , . . . , m } → { , . . . , m } be the permutation that “orders” the columns according to the subsets,while maintaining the order inside each subset, that is, p ( i ) = 1 ,p ( i ) = 2 , ... p ( i | J | + ··· + | J b | ) = m. This definition implies p ( J k ) is the set of consecutive integers | J | + · · · + | J k − | + 1 .. | J | + · · · + | J k | . Let Q bethe permutation matrix that embodies the column interchanges defined by p : Q . j = e i j , for j = 1 , . . . , m, where the m -column vector e i j is zero except for its i j -th entry, which is one. Then( Q − N Q ) rc = ( Q T N Q ) rc = e Ti r N e i c = n i r i c , which implies ( Q − N Q ) p ( J k ) p ( J k ) = N J k J k . Thus Q has the effect of grouping entries in each block. Furthermore, if i r and i c belong to the same subset inthe column/row partition, say J k , then i r < i c (resp., i r = i c ) if and only if p ( i r ) < p ( i c ) (resp., p ( i r ) = p ( i c )),so Q T N Q has a block diagonal structure, with upper triangular diagonal blocks ( Q T N Q ) p ( J k ) p ( J k ) , k = 1 , . . . , b ,each with nonzero diagonal elements. This implies that the inverse of Q T N Q has the same structure, where eachdiagonal block is replaced by its inverse. Entries of ( Q T N Q ) − outside these blocks are zero, therefore entriesin N − outside ∪ bk =1 J k − × J k are zero. In particular, this implies that, for k = 0 , . . . , b ,( N J k J k ) − = (( Q T N Q ) p ( J k ) p ( J k ) ) − = ( Q T N Q ) − p ( J k ) p ( J k ) = ( Q T N − Q ) p ( J k ) p ( J k ) = ( N − ) J k J k . Additionally, we have that, ( N − ) II = ( Q T N Q ) − p ( I ) p ( I ) . Q T N Q implies that ( Q T N Q ) p ( I ) ,p ( J k ) = 0. Let L = J k ∩ { j − } . Then( Q T N Q ) − p ( I ) p ( L ) = 0 because the block ( Q T N Q ) − p ( J k ) p ( J k ) is upper triangular. The result( N − ) II = 0follows from the fact I = L ∪ J k . The remaining result follows analogously. (cid:3) Proposition 4
Let (∆ , ∆ , . . . ) and ( P , P , . . . ) be the sequence of connection and change-of-basis matrices,respectively, produced by the application of the Sweeping Algorithm over Z to the connection matrix ∆ ∈ Z m × m with column/row partition J , . . . , J b . Then, for r ≥ , we have(i) ∆ r and ∆ P r − is compliant with the allowable sparsity pattern of ∆ ;(ii) the nonzero entries of ∆ P r − strictly below the r -th diagonal are located on or above a unique primarypivot position, entries in primary pivot positions are nonzero;(iii) the nonzero entries of ∆ r strictly below the r -th diagonal are either primary pivots (always nonzero) or areabove a unique primary pivot;(iv) each linear integer program formulated in the matrix P r − computation step has an optimal solution;(v) the change-of-basis matrices P r − have the following upper triangular structure: the diagonal elementsare nonzero, the positions of the off-diagonal nonzero elements are contained in ( ∪ bi =1 J i × J i ) ∩ { ( i, j ) ∈{ , . . . , m } | i < j } . Proof . For r = 1, items (i)–(v) are either trivially true, since P = I , which implies ∆ = ∆ = ∆, whosediagonal is zero, or vacuously true. The proof is by induction.Assume by induction that the items are true for fixed arbitrary r , greater than 1. First we show that(iv) and (v) are satisfied for r + 1. Consider the sweeping of the r -th diagonal and let the entry in position( j − r, j ) be marked as a change-of-basis pivot. Let k be the chain index associated with column j . Let I = J k − ∩ { j − r, . . . , m } , J = J k ∩ { , . . . , j } , c = | J | . Using Lemma 3 and the facts that the inductionhypothesis that (v) is satisfied for r , we have that∆ rIJ = ( P r − ) − I . ∆ P r − . J = ( P r − ) − II ∆ IJ P r − JJ . We will construct a column vector y ∈ R c such that ∆ rIJ y = 0. Let p be the column containing the primarypivot entry to the left of ∆ rj − r,j . Then, by induction hypothesis (i), j − r ∈ I and p ∈ J . Additionally, byinduction hypothesis (iii), the entries below row j − r in columns p and j of ∆ r are zero. Now let y be a zerovector except for the entries corresponding to columns ∆ rIp , whose value is − ∆ rj − r,j , and ∆ rIj (the c -th, or last,entry of y ), whose value is ∆ rj − r,p . Then ∆ rIJ y = 0 . The integrality of P r − and ∆ imply that y is rational, and thus also P r − JJ y . The facts that P r − JJ is uppertriangular with nonzero diagonal entries and that y c (cid:54) = 0 imply that ( P r − JJ y ) c (cid:54) = 0. Thus there is a suitablemultiple ¯ x of y that is an integral solution to (cid:40) ( P r − ) − II ∆ IJ x = 0 ,x c ≥ , (cid:40) ∆ IJ x = 0 ,x c ≥ . Therefore the linear integral problem in the matrix P r computation step is feasible. Its objective is also boundedby construction, since x c ≥
1. We may conclude, see [10], that this linear integral problem has an optimalsolution x ∗ . Hence (iv) is for P r . Furthermore, the facts that I ⊂ J k − and J ⊂ J k , and the assignment rule P rJj ← x ∗ imply that (v) is satisfied for P r .Now consider (i) and (ii) for ∆ P r . By induction, (i) and (ii) are satisfied for ∆ P r − . By constructionof P r , the only columns that change from ∆ P r − to ∆ P r are the ones containing change-of-basis entriesin ∆ r . The corresponding columns in P r are built via the solution of the linear programs in the matrix P r computation step. This guarantees that the entries on and below the change-of-basis position in the product∆ P r are zero. Since the change-of-basis position is on the r -th diagonal, it is strictly below the ( r + 1)-thdiagonal, so (ii) is satisfied for these columns. Also, since we are replacing the column containing the change-of-basis entry with a linear combination of this column and other columns to its left, all with same chainindex, the allowable sparsity pattern of ∆ is preserved, so (i) is also satisfied for these columns. Now considera column s that was not changed. Trivially, (i) is satisfied in this case. To establish (ii), we consider thethree possibilities for (∆ P r ) { s − r,...,m } ,s = (∆ P r − ) { s − r,...,m } ,s . If (∆ P r − ) { s − r +1 ,...,m } ,s (cid:54) = 0, (ii) is valid byinduction. If (∆ P r − ) { s − r,...,m } ,s = 0, (ii) is trivially satisfied. The remaining possibility is (∆ P r − ) s − r,s (cid:54) = 0and (∆ P r − ) { s − r +1 ,...,m } ,s = 0. In this case, using induction and (v),∆ rs − r,s = ( P r − ) − s − r, { s − r,...,m } (∆ P r − ) { s − r,...,m } ,s = ( P r − ) − s − r,s − r (∆ P r − ) s − r,s (cid:54) = 0 , whereas, for i > s − r , ∆ ri,s = ( P r − ) − i, { i,...,m } (∆ P r − ) { i,...,m } ,s = 0 . But these facts imply that ∆ rs − r,s must have been marked as a primary pivot when the r -th diagonal was swept.Thus the unique nonzero element of ∆ P r in column s , strictly below the ( r + 1)-th diagonal, is located on aprimary pivot position and (ii) is satisfied.Next we show (i) and (iii) are true for ∆ r +1 . Given the (established above) upper triangular structure of P r ,shared by its inverse, if i ∈ J k , then ∆ r +1 i . = (cid:88) i (cid:48) ∈ J k ∩{ i +1 ,...,m } P ri,i (cid:48) (∆ P r ) i (cid:48) . . Then the fact that ∆ P r satisfies (i), imply ∆ r +1 also satisfies. Furthermore, the fact that ∆ P r satisfies (ii)implies that the nonzero entries strictly below the ( r + 1)-th diagonal are located on or above a primary pivotposition. Now suppose position ( i, (cid:96) ), strictly below the ( r + 1)-th diagonal, is marked as a primary pivot. Then,using (ii), ∆ r +1 i(cid:96) = (cid:88) i (cid:48) ∈ J k ∩{ i +1 ,...,m } P ri i (cid:48) (∆ P r ) i (cid:48) (cid:96) = P ri i (∆ P r ) i(cid:96) (cid:54) = 0 , so the primary pivot entries in ∆ r +1 are nonzero. Finally, algorithm rules allow at most one primary pivot percolumn, so the uniqueness in (iii) is easily guaranteed. (cid:3) The following corollary addresses the configuration of the last matrix obtained by the Sweeping Algorithmover Z and is a simple consequence of Proposition 4. 10 orollary 5 Let ∆ m be the last matrix produced by the application of the Sweeping Algorithm over Z to theconnection matrix ∆ ∈ Z m × m . Then the primary pivot entries are nonzero and each nonzero entry is locatedabove a unique primary pivot. As in the special case of grouped connection, see [5], the configuration established in Corollary 5 of thelast connection matrix in the sequence produced by the algorithm leads to the complementary relation betweencolumns and rows expressed in the next proposition. The proof is a straightforward adaption of the proof in [5].
Proposition 6
Let ∆ m be the last matrix produced by the application of the Sweeping Algorithm over Z to theconnection matrix ∆ ∈ Z m × m . If the j -th column of ∆ m is nonzero, then its j -th row is null, or, equivalently, ∆ m . j ∆ mj . = 0 , for all j. (1) Proof . Equation (1) is trivial when ∆ m . j is a zero column. So suppose ∆ m . j (cid:54) = 0. By the inherited allowable sparsitypattern established in Proposition 4, there exists s such that j ∈ J s . By Corollary 5, the nonzero columns of ∆ m are precisely the columns containing primary pivots. Label the primary pivots of columns in J s in increasingorder of row index: if ∆ mi j , . . . , ∆ mi a j a are the primary pivots in columns in J s , then i < i < · · · < i a . Thus, j , j , . . . , j a are the nonzero columns in ∆ m . J s and j ∈ { j , . . . , j a } . Furthermore, ∆ mi a j a is the unique nonzeroentry of row ∆ mi a . , row i a − has a nonzero entry in column j a − and may have another one in column j a , and soon. The fact that ∆ m is nilpotent implies that0 = ∆ mi a . ∆ m . j (cid:48) = ∆ mi a j a ∆ mj a j (cid:48) , for fixed arbitrary j (cid:48) . Using the fact that the primary pivot entry is nonzero, we conclude that ∆ mj a j (cid:48) = 0, forall j (cid:48) . Repeating the argument for i a − and using the fact that the j a -th row of ∆ m is null, we establish thatits j a − -th row is null. The nullity of rows j a − , . . . , j of ∆ m follow analogously. Therefore, we conclude that∆ m . j = 0. (cid:3) If the connection matrix has entries in a field F , the sweeping algorithm can be easily adapted by letting thevariables in the minimization problem in the Matrix P r computation step have entries in F , and not in Z . Letus call this version the Accumulated Sweeping Algorithm over F . The adjective “accumulated” refers to matrix P r , which accumulates all information regarding the successive basis changes along iterations 1, . . . , r . Theadaptation of Proposition 4 to this new algorithm is presented below. Proposition 7
Let (∆ , ∆ , . . . ) and ( P , P , . . . ) be the sequence of connection and change-of-basis matrices,respectively, produced by the application of the Accumulated Sweeping Algorithm over F to the connection matrix ∆ ∈ F m × m with column/row partition J , . . . , J b . Then, for r ≥ , we have(i) ∆ r and ∆ P r − is compliant with the allowable sparsity pattern of ∆ ;(ii) the nonzero entries of ∆ P r − strictly below the r -th diagonal are located on or above a unique primarypivot position, entries in primary pivot positions are nonzero;(iii) the nonzero entries of ∆ r strictly below the r -th diagonal are either primary pivots (always nonzero) or areabove a unique primary pivot; iv) the optimization program formulated in the matrix P r computation step corresponding to change-of-basispivot ∆ rj − r,j , where j ∈ J k , has an optimal solution x = P r − JJ y , where y ∈ F c is given by y s = − ∆ rj − r,j ∆ rj − r,p , if s = p (cid:48) , , if s = c, , otherwise, (2) where p (cid:48) is the column of ∆ rIJ that contains ∆ rj − r,p , the primary pivot to the left of the change-of-basispivot ∆ rj − r,j ;(v) the change-of-basis matrices P r − have the following upper triangular structure: the diagonal elementsare equal to , the positions of the remaining nonzero elements are contained in ( ∪ bi =1 J i × J i ) ∩ { ( i, j ) ∈{ , . . . , m } | i < j } . Proof . The only items that need to be proved are (iv) and (v). For the remaining ones, the proof of Proposition4 is valid.When r = 1 item (iv) is vacuously true and (v) is trivially true, since P is the identity matrix. Assume theyare true for fixed arbitrary r , greater than 1. Consider the case r + 1, that is, matrix P r will be constructedbased on the sweeping of the r -th diagonal of ∆ r . Suppose ∆ rj − r,j is a change-of-basis pivot and ∆ rj − r,p is theprimary pivot entry to its left. Suppose j ∈ J k and let I and J be as defined in the algorithm. Let p (cid:48) be thecolumn of ∆ rIJ that contains the primary pivot entry ∆ rj − r,p . By (iii) and the rules of the algorithm, we have(∆ rIJ ) . p (cid:48) = ∆ rj − r,p and (∆ rIJ ) . c = ∆ rj − r,j . Furthermore, by the induction hypotheses, the vector y in (iv) is well defined and belongs to F c . Therefore∆ rIJ y = 0 . Let x = P r − JJ y. Since ∆ rIJ = ( P r − ) − II ∆ IJ P r − JJ , we have that x satisfies ∆ IJ x = 0. Furthermore, by theinduction hypothesis (v), x c = ( P r − JJ ) c . y = y c = 1 , which proves (iv) and (v), given that P r is constructed from P r − by the replacement P rJj = x, for each change-of-basis column j . (cid:3) Proposition 7 implies we can simplify the matrix P r computation step in the Accumulated Sweeping Algo-rithm over F , by using the ready-made optimal solution x = P r − JJ y as described in its item (iv). Now noticethat, if x = P r − JJ y , for y described in item (iv) of the proposition, then P rJj = − ∆ rj − r,j ∆ rj − r,p P r − Jp + P r − Jj . Using simple algebra, we have that P rJJ = P r − JJ I · · · y . P r − . p and P r − . j outside rows in J are zero, all updatesrelative to the sweeping of the r -th diagonal may be represented by one single matrix T r , that coincides withthe identity matrix, except for columns corresponding to change-of-basis columns. For each such column j , if∆ rj − r,p is the primary pivot to the left of the change-of-basis pivot ∆ rj − r,j , the matrix T r contains the entry − ∆ rj − r,j / ∆ rj − r,p in position ( p, j ). Summarizing, this matrix is given by T rij = , if i = j, − ∆ rj − r,j ∆ rj − r,p , if i = p, ∆ rj − r,j is a change-of-basis pivot and∆ rj − r,p is the primary pivot to its left,0 , ow. (3)With this definition, the accumulated change-of-basis P r satisfies P r = P r − T r . (4)Given that ∆ r +1 = ( P r ) − ∆ P r = ( P r − T r ) − ∆ P r − T r = ( T r ) − ∆ r T r , (5)the algorithm may be simplified by dealing exclusively with the T r transition matrices , thus avoiding the productin (4). To distinguish this simplified version from the previous one, we name it the Incremental SweepingAlgorithm over F , in the sense that in this case we keep a matrix that performs an incremental change, from∆ r to ∆ r +1 , say, instead of a matrix that accumulates all changes, taking ∆ directly to ∆ r +1 . This algorithm,inserted below for completeness, was introduced in [5]. Although it was defined and studied with groupedconnection matrices in mind, we’ve seen that it may be equally applied to general connection matrices. Inparticular, the items of Proposition 7 that concern the sequence ∆ , ∆ , . . . , are satisfied by the IncrementalSweeping Algorithm over F .The incremental way of updating the connection matrix is just a computational detail. The relevant differ-ence between the accumulated and the incremental algorithms is that, in the former, any optimal solution to theoptimization problem may be used, while, in the latter, a specific optimal solution is employed. Incremental Sweeping Algorithm over F Input: nilpotent upper triangular matrix ∆ ∈ F m × m with column/row partition J , J , . . . , J b . Initialization Step: r = 0∆ r = ∆ T r = I ( m × m identity matrix) Iterative Step: (Repeated until all diagonals parallel and to the right of the main diagonal have been swept)
Matrix ∆ update r ← r + 1∆ r = ( T r − ) − ∆ r − T r − Markup
Sweep entries of ∆ r in the r -th diagonal: If ∆ rj − r,j (cid:54) = 0 and ∆ r . ,j does not contain a primary pivot Then If ∆ rj − r . contains a primary pivot Then temporarily mark ∆ rj − r,j as a change-of-basis pivot Else permanently mark ∆ rj − r,j as a primary pivot Matrix T r construction T r ← I For each change-of-basis pivot ∆ rj − r,j , change the j -th column of T r as followsLet p be such that ∆ rj − r,p is a primary pivot T rpj ← − ∆ rj − r,j / ∆ rj − r,p Final Step:
Matrix ∆ update r ← r + 1∆ r = ( T r − ) − ∆ r − T r − Output: (∆ , . . . , ∆ m ) and ( T , . . . , T m − )Let P r be the set of column indices of primary pivots of ∆ m on or below the r -th diagonal. Then P = ∅ and P r \P r − is the set of column indices of primary pivots on the r -th diagonal, for r = 1 , . . . , m −
1. Proposition 7,the definition of T r given in (3), the relationship between the change-of-basis matrices given in (4) and thedefinition of the Incremental Sweeping Algorithm over F imply the following result. Proposition 8
Let (∆ (= ∆) , ∆ , . . . ) and ( T , T , . . . ) be the sequence of connection and change-of-basis ma-trices produced by the Incremental Sweeping Algorithm over F , where ∆ is a connection matrix in F m × m , withcolumn/row partition J , . . . , J b . Then, for r = 1 , . . . , m , we have(i) ∆ r is compliant with the allowable sparsity pattern of ∆ ;(ii) the nonzero entries of ∆ r strictly below the r -th diagonal are either primary pivots or are above a uniqueprimary pivot;(iii) primary pivot entries of ∆ r are nonzero;(iv) the transition matrices T r have the following upper triangular structure: the diagonal elements are equal to , each change-of-basis column has two nonzero entries and the positions of the off-diagonal nonzero ele-ments are contained in ( ∪ bi =1 (( P r − \P r − ) ∩ J i ) × (( P r − \P r − ) ∩ J i ) ∩ { ( i, j ) ∈ { , . . . , m } | i < j } . The next corollary and proposition follow from Proposition 7(ii) and mimic analogous results for the SweepingAlgorithm over Z . Corollary 9
Let ∆ m be the last matrix produced by the application of the Incremental Sweeping Algorithm over F to the connection matrix ∆ ∈ F m × m . Then the primary pivot entries are nonzero and each nonzero entry islocated above a unique primary pivot. Temporary marks are erased at the end of the iterative step. roposition 10 Let ∆ m be the last matrix produced by the application of the Incremental Sweeping Algorithmover F to the connection matrix ∆ ∈ F m × m . If the j -th column of ∆ m is nonzero, then its j -th row is null, or,equivalently, ∆ m . j ∆ mj . = 0 , for all j. (6)The change of basis that effects the transition from ∆ r to ∆ r +1 = ( T r ) − ∆ r T r is constructed so as to zeroout in ∆ r +1 the change-of-basis entries, without disturbing the (already null) entries below the change-of-basispivots. If { j , . . . , j t r } denotes the set of distinct column indices that contain change-of-basis entries in ∆ r and { p , . . . , p t r } the column indices such that ( j s − r, p s ) is the position of the primary pivot in the same row as thechange-of basis pivot ∆ rj s − r,j s , for s ∈ { , . . . , t r } , then the matrix constructed in the iterative step, that effectsthis basis change, may be expressed as T r = I − t r (cid:88) s =1 ∆ rj s − r,j s ∆ rj s − r,p s U p s j s = I − t r (cid:88) s =1 α p s j s U p s j s , (7)where U ij is the m × m matrix unit whose entries are all 0 except in position ( i, j ), where it is 1. These matrixunits have the property, see [9], U ij U k(cid:96) = δ jk U i(cid:96) . (8)Algebraically, the post-multiplication of ∆ r by T r in (7) consists of t r elementary column operations on thechange-of-basis columns of ∆ r . Of the three possible elementary column operations on a column j , we use onlyone in this work: “add to column j a multiple of another column”. The property (8) and the fact that no columnof ∆ r may contain both a primary pivot mark and a change-of-basis pivot imply that T r = ( I − α p j U p j ) · · · ( I − α p tr j tr U p tr j tr ) , (9)and the order of the matrix product on the right-hand-side of (9) is irrelevant. Furthermore, as already noted in[5], the inverse of T r is given by( T r ) − = I + t (cid:88) s =1 α p s j s U p s j s = ( I + α p j U p j ) · · · ( I + α p tr j tr U p tr j tr ) . (10)This follows from (7), (8) and the fact that a column cannot simultaneously contain a primary pivot and achange-of-basis pivot. In keeping with its counterpart, the pre-multiplication of ∆ r T r by ( T r ) − consists of t r elementary row operations. Amongst the ones available, the only row operation on row i considered herein is ofthe type “add to row i a multiple of another row”. This discussion implies the following lemma. Lemma 11
Suppose the Incremental Sweeping Algorithm over F is applied to the connection matrix ∆ ∈ F m × m ,with column/row partition J , . . . , J b . Then the update from ∆ r to ∆ r +1 may be accomplished blockwise,according to the individual updates ∆ rJ k − J k = ( T r − J k − J k − ) − ∆ r − J k − J k T r − J k J k , for k = 1 , . . . , b. (11) Consequently, only columns containing change-of-basis pivots are subjected to elementary column operations, andonly rows with the same index as the column of the primary pivots used for canceling out the change-of-basispivots suffer elementary row operations. roof . Items (i) and (iv) of Proposition 8 imply that the connection matrix update step may be restricted tothe entries in ∪ bk =1 J k − × J k , since entries outside these positions are zero. The update for the individual blocksin ( ?? ) is obtained using Lemma 3. Column operations are due to the post-multiplication, and by constructionof T r − J k J k , affect only columns of J k that contain change-of-basis pivots. Using (10), the pre-multiplication by( T r − J k − J k − ) − affects only rows with same index as the columns that contain the primary pivots used for cancelingout the change-of-basis pivots. (cid:3) Let ∆ ∈ F m × m be a connection matrix, with column/row partition J , . . . , J b , to which the IncrementalSweeping Algorithm over F is applied. Suppose there is a change in entry in position ( i, j ) ∈ J k − × J k from∆ r to ∆ r +1 . Lemma 11 implies this may be due to an elementary column operation on column j , due to achange-of-basis mark on entry ( j − r, j ) on this column, and/or to an elementary row operation, due to a primarypivot in column i ∈ J k − that is being used to cancel out a change-of-basis pivot in position ( q − r, q ) in somecolumn q ∈ J k − , where q > i . Let J k be the set of columns in J k that contain primary pivot entries in ∆ m ,for k = 1 , . . . b . Let J k = J k \J k , for all k . Corollary 9 and Proposition 10 imply that rows of ∆ m in J k mustbe zero, for all k , and thus cannot contain primary pivots. Furthermore, since change-of-basis pivots must belocated to the right of and on the same row as primary pivots, there can be no change-of-basis pivots on rows in J k , for all k . Putting together these facts leads us to the last ingredient needed to reformulate the IncrementalSweeping Algorithm over F . Lemma 12
Let ∆ ∈ F m × m be a connection matrix with column/row partition J , . . . , J b . Consider the sweepingof the r -th diagonal of ∆ r in the Incremental Sweeping Algorithm over F . The markings on entries in columnsbelonging to J k and the construction of T rJ k J k are completely determined by the values of the change-of-basis andprimary pivots in ∆ rJ k − J k . Proof . Consider a fixed generic entry ∆ rj − r,j , where j ∈ J k , eligible for receiving a mark. Since zero entries arenot marked, we may suppose ∆ rj − r,j (cid:54) = 0 and, using Proposition 8 (i), conclude that j − r ∈ J k − . Furthermore,rows in J k − cannot contain primary or change-of-basis pivots, so we may assume j − r ∈ J k − .Nonzero entry ∆ rj − r,j will not be marked if there exists a primary pivot below it, say in position ( i, j ), i > j − r , which would imply that i ∈ J k − , which means this primary pivot is in ∆ rJ k − J k .Now suppose column j does not contain a primary pivot. If ∆ rj − r . contains a primary pivot, say in position( j − r, p ), then, Proposition 8 (iii), ∆ rj − r,p (cid:54) = 0, and, by item (i), p ∈ J k , so this primary pivot is located in∆ rJ k − J k . In this case, ∆ rj − r,j will be marked as a change-of-basis pivot. Otherwise, it will be marked as aprimary pivot. Notice that, in this case j − r ∈ J k − , since rows in J k − cannot contain primary pivots.Consider the construction of T rJ k J k . The diagonal is always initialized with ones. Column T r . j will have anothernonzero entry only if ∆ rj − r,j is a change-of-basis pivot. Suppose this is the case. Then, as we’ve establishedabove, for some p ∈ J k , there is a primary pivot ∆ rj − r,p , with j − r ∈ J k − and p ∈ J k . This will give rise to thenonzero entry − ∆ rj − r,j / ∆ rj − r,p in position ( p, j ) of T r . (cid:3) Lemma 12 implies the sweeping could be done sequentially blockwise as follows. Let ∆ ∈ F m × m be a con-nection matrix with column/row partition J , . . . , J b . During the blockwise sequential procedure, all connectionmatrices have the same column/row partition as ∆. At the first step, one would apply the sweeping algorithm toa connection matrix that coincides with ∆ in positions J × J , and is zero otherwise. The transition submatrices T ,rJ J , for r = 1, . . . , m , produced during the execution of the algorithm are recorded and will be used in the next16tep. At step k , for k = 1, . . . , b , we consider a connection matrix that coincides with ∆ in positions J k − × J k ,and is zero otherwise. The diagonals of this matrix are swept as in the Incremental Sweeping Algorithm over F , and the same rules are used for marking up the entries, as well as building the transition matrix, which willcontain nonzero off-diagonal entries only in positions J k × J k , by virtue of the construction of the connectionmatrix used as input. The remaining principal submatrices T k,rJ s J s , for s (cid:54) = k , are identity matrices, for all r . Inthe update step, only entries in positions J k − × J k need be updated. The block update formula (11) is employed,with the difference that the submatrices T k − ,rJ k − J k − produced in the previous step are used, instead of the onesproduced during the markup in the current step. Lemma 12 implies that submatrix T k,rJ k J k produced in step k of this sequential procedure coincide with submatrix T rj k J k constructed in the application of the IncrementalSweeping Algorithm over F to ∆. Therefore the evolution of entries in positions J k − × J k in step k coincideswith the evolution of entries of ∆ in the same positions, during the application of the algorithm to ∆. Thesequence ∆ , . . . , ∆ m can be obtained summing the sequences produced by sweeping the 1-block matrices asdescribed.Nevertheless, the procedure described above is still not simple enough for our purposes. Our aim is to forgothe pre-multiplication operation, and that of course comes at a price. Proposition 10 and Lemma 11 imply thatthe only rows that change in the pre-multiplication are the ones that are equal to zero at the end of the sweeping.Furthermore, Lemma 12 implies that the entries in these rows do not contribute to the blocks T k,rJ k J k . Therefore,the sequential sweeping can be even further simplified, if we give up keeping track of the evolution of the rowsin ∪ bk =1 J k . Instead of storing the blocks T k − ,rJ k − J k − , one need only record the primary pivot columns of the lastconnection matrix in step k − F , since allblocks of connection matrix ∆( k ) are zero, except for block ∆( k ) J k − ,J k , for k = 1 , . . . , b . Block Sequential Sweeping Algorithm over F Input: nilpotent m × m upper triangular matrix ∆ with column/row partition J , J , . . . , J b . Initialization Step: J = ∅ Iterative Step:For k = 1 , . . . , b do Let ∆( k ) be the matrix obtained from ∆ by zeroing rows in J k − and entriesoutside positions in J k − × J k Apply the Incremental Sweeping Algorithm over F to ∆( k ), with column/rowpartition J , . . . , J b , obtaining ∆( k ) m J k = indices of columns of ∆( k ) m containing primary pivots Output: (∆( k ) , . . . , ∆( k ) m ) and ( T ( k ) , . . . , T ( k ) m − ), for k = 1 , . . . , b Theorem 13 (Uncoupling)
Let ∆ ∈ F m × m be a connection matrix with row/column partition J , . . . , J b .Let ∆ m be the matrix produced by the Incremental Sweeping Algorithm over F applied to ∆ , and let ∆( k ) m , for k = 1 , . . . , b , be the matrices obtained in the Block Sequential Sweeping Algorithm over F applied to ∆ . Then ∆ mJ k − J k = ∆( k ) mJ k − J k , for k = 1 , . . . , b and the collection of change-of-basis and primary pivots encountered n the application of the Incremental Sweeping Algorithm over F to ∆( k ) , for k = 1 , . . . , b , coincides with thechange-of-basis and primary pivots found when it is applied to ∆ . Proof . Let T ( k ) , T ( k ) , . . . , T ( k ) m be the transition matrices constructed when applying the IncrementalSweeping Algorithm over F to ∆( k ). Given that ∆( k ) has at most one nonzero block, ∆( k ) J k − J k , Proposition 8,Lemma 3 and induction imply that T ( k ) rJ i J i is an identity matrix, for all i (cid:54) = k . Now, using Lemma 11, theupdate of ∆( k ) is reduced to the following update∆( k ) rJ k − J k = ( T ( k ) r − J k − J k − ) − ∆( k ) r − J k − J k T ( k ) r − J k J k = ∆( k ) r − J k − J k T ( k ) r − J k J k , for all r ≥
1. Since the update of ∆( k ) involves only the post-multiplication step, only elementary columnoperations are performed.Lemma 12 implies that ∆(1) mJ J = ∆ mJ J and the change-of-basis and primary pivots marked during theapplication of the algorithm to ∆(1) coincide with the ones marked in columns in J when the algorithm isapplied to ∆. Assume by induction that ∆( k − mJ k − J k − = ∆ mJ k − J k − and change-of-basis and primary pivotsof ∆( k −
1) agree with the ones in columns in J k − marked when ∆ is swept. Also by induction, J k − = N k − .Proposition 10 implies rows of ∆ m in J k − are zero. By construction, rows of ∆( k ) in J k − are also zero, and,since ∆( k ) suffers only elementary column operations during the application of the Incremental Sweeping over F , these rows are not changed. So ∆ m J k − J k = ∆( k ) m J k − J k .By Lemma 11, entries in the rows of ∆ in J k − are not subjected to elementary row operations duringthe application of the Incremental Sweeping Algorithm over F thereto. Furthermore, all change-of-basis andprimary pivots in columns in J k occur in positions in J k − × J k . Hence changes to entries in these rows areonly due to elementary column operations, as also happens when the Incremental Sweeping is applied to ∆( k ).Since ∆ J k − J k = ∆( k ) J k − J k and the elementary column operations on columns in J k are solely dependenton the entries in this submatrix, it follows that the change-of-basis and primary pivots marked in columnsin J k during execution of the Incremental Sweeping Algorithm over F to both ∆ and ∆( k ) coincide, and so∆ mJ k − J k = ∆( k ) mJ k − J k . By induction, the statements are true for k = 1 , . . . , b . (cid:3) The Uncoupling Theorem 13 will lead to a significant simplification of the task of studying the effects of theapplication of the Incremental Sweeping Algorithm over F to a connection matrix with row/column partition J ,. . . , J b , as long as one is content to forgo the observation of the evolution of rows in ∪ bk =1 J k , which are known toend up zero and will not contain neither primary nor change-of-basis pivots. To ease the discussion that follows,we henceforth call the special case of the Incremental Sweeping Algorithm over F , where the input is restricted toconnection matrices with row/column partition consisting of only two subsets, the F .The analysis of the 1-Block Incremental Sweeping Algorithm over F is much easier than that of its moregeneral counterpart. If the columns of the connection matrix ∆ ∈ F m are partitioned into two subsets J and J , only rows in J are altered in the pre-multiplication by ( T r − ) − , in the matrix update step. But ∆ J = 0and this zero pattern is invariant under elementary column operations. So the post-multiplication part of theupdate doesn’t change the nullity of rows in J and the elementary row operations performed during the pre-multiplication part of the update step involve only the zero rows in J . This implies ∆ rJ = 0 for all r , and wemay eliminate the pre-multiplication part of the update step, so that only elementary column operations needbe performed. 18uring the execution of the 1-Block Incremental Sweeping Algorithm over F , columns of ∆ r may be classifiedas active (resp., passive ), if they contain (resp., do not contain) a primary pivot mark. The active columnseffect change upon the passive columns. The passive columns suffer changes caused by active columns. At thebeginning of the algorithm all columns are passive and before changes are allowed to happen, at least one columnmust become active. Once a column reaches the active state, it doesn’t leave it, since primary pivot marks arepermanent. Passive columns undergo a (possibly empty) sequence of elementary column operations and eitherreach an active state or become zero, since Corollary 9 implies columns without primary pivots must be zero. Ifa column reaches an active state, it does so when the lowest nonzero entry in the column is marked as a primarypivot. The order of sweeping implies that the change-of-basis pivots that occur in a fixed column, say j , aremarked in an upward fashion. If the entry in position ( j − r, j ) is marked as a change-of-basis pivot, and theentry in position ( j − r, p ) contains the primary pivot to its left, columns p and j exhibit a sequence of trailing ofzeros from row j − r + 1 to the last row. The elementary column operation that eliminates this change-of-basispivot changes only the entries in rows 1 through j − r , the actual operation being determined by the values ofthe two pivots. By construction, each operation increases the number of trailing zeros by at least one.In the 1-Block Incremental Sweeping Algorithm over F , the passive columns dictate the cancellations, whichare done only once a change-of-basis pivot is marked. We propose a reengineered version therefor, in which thisrole is transferred to the active columns. Once an primary pivot is identified, all cancellations it is responsiblefor in the 1-block Incremental Sweeping Algorithm over F are performed. To arrive at the same final matrix asin the original algorithm, the primary pivots must also be identified in a upward order, from the bottom up.The second and last important aspect for the identification of primary pivots, is the left-to-right order of thesweeping. The algorithm below incorporates both of these. In the algorithm we adopt the usual convention thatif S is an empty set and A is a real matrix, then A S . = 0. Since we are considering connection matrices withat most one nonzero block, we may assume, without loss of generality, that the row/column partition has twosubsets. Revised 1-Block Incremental Sweeping Algorithm over F Input: nilpotent m × m upper triangular matrix ∆ with column/row partition J , J . Initialization Step: C = { , . . . , m } , ˜∆ = ∆, t = 1. Iterative Step:While ˜∆ t . C t (cid:54) = 0 do Let i t = max { i | ˜∆ tiC t (cid:54) = 0 } Let j t = min { j ∈ C t | ˜∆ ti t j (cid:54) = 0 } Permanently mark ∆ ti t j t as a primary pivot Update Matrix Construction ˜ T t ← I − (cid:88) j ∈ C t j > j t ˜∆ ti t j ˜∆ ti t j t U j t j (cid:34) Simplified Matrix ∆ update ˜∆ t +1 = ˜∆ t ˜ T t C t +1 ← C t \{ j t } t ← t + 1 19 utput: ( ˜∆ , . . . ) and ( ˜ T , . . . )The attentive reader may realize that the Revised 1-block Incremental Sweeping Algorithm over F is animplementation of the transposed version of the Gaussian method for obtaining a row echelon matrix, minus therow swaps (which would amount to column swaps in the revised algorithm). Proposition 14
Let ( i , j ) , . . . , ( i t ∗ , j t ∗ ) be the positions of primary pivots marked in the application of theRevised 1-block Incremental Sweeping Algorithm over F to the connection matrix ∆ ∈ F m × m with row/columnpartition J , J , in the order in which they were marked. Then the following are true:(i) once a column receives a primary pivot mark, it remains invariant until the end of the algorithm,(ii) j t > i t , for t = 1 , . . . , t ∗ ,(iii) i > i > · · · > i t ∗ ,(iv) ˜∆ t +1 { i t ,...,m } C t +1 = 0 , for t = 1 , . . . , t ∗ ,(v) the number of consecutive zero entries at the bottom of each column never decreases. Proof . Without loss of generality, suppose ∆ (cid:54) = 0, otherwise the proposition is vacuously true. Given theconstruction of ˜ T t , only columns j such that j > j t and j ∈ C t change from ˜∆ t to ˜∆ t +1 . Namely,˜∆ t +1 . j = ˜∆ t . j − ˜∆ titj ˜∆ titjt ˜∆ t . j t , if j ∈ C t and j > j t , ˜∆ t . j , otherwise , (12)so (i) is true.The initial matrix is upper triangular and this property is preserved under the update step, since columnsthat changed receive the sum of the old column and a multiple of a column to its left. Hence j t > i t for all t ,since ˜∆ ti t j t (cid:54) = 0. Thus (ii) is also true.By construction, the lowest row of ˜∆ with nonzero entries is i . Formula (12) gives˜∆ i j = ˜∆ i j − ˜∆ i j ˜∆ i j ˜∆ i j = 0 , if j > j , ˜∆ i j , if j ≤ j . (13)Using the fact that there are no nonzero entries to the left of ˜∆ i j , the only nonzero entry on or below row i in ˜∆ is the primary pivot in position ( i , j ), which proves (iv) for t = 1. Since j / ∈ C , the lowest nonzero rowof ˜∆ . C must be higher than i , so i > i .Assume by induction that i > · · · > i t − and ˜∆ t { i t − ,...,m } C t = 0. The last fact implies i t − > i t , proving(iii) by induction. The definition of j t implies ˜∆ t { i t +1 ,...,m } C t = 0. Let j / ∈ { j , . . . , j t } . These zero entrieswon’t be altered by the elementary column operations embodied in the matrix update formula (12), hence˜∆ t +1 { i t +1 ,...,m } C t = 0. When i = i t , (12) implies˜∆ t +1 i t j = ˜∆ ti t j − ˜∆ titj ˜∆ titjt ˜∆ ti t j t = 0 , if j ∈ C t and j > j t , ˜∆ ti t j , if j ∈ C t and j ≤ j t . Using the that ( i t , j t ) is the position of the leftmost nonzero entry of ˜∆ ti t C t , we conclude that ˜∆ t +1 i t j t is the onlynonzero entry on row i t in the submatrix ˜∆ t +1 . C t +1 . Therefore (iv) is true for ˜∆ t +1 . By induction, it follows that(iv) is true for all values of t . 20inally, to show (v), notice that a column j suffers change when j ∈ C t \{ j t } , for some t . But if this is so,the entries in column j , in rows i t + 1 , . . . , m must be zero and the update (12) will zero the entry in position( i t , j ) while keeping the entries lower down at zero. Entries higher up may or not zero out, and may even changefrom zero to a nonzero value. But if a column changes, the number of trailing zeros in that column will alwaysincrease by at least one. (cid:3) Corollary 15
Let ˜∆ t ∗ +1 be the last matrix obtained by the application of the Revised 1-block Incremental Sweep-ing Algorithm to the connection matrix ∆ ∈ F m × m with row/column partition J , J . Then the primary pivotentries are nonzero and each nonzero entry of ˜∆ t ∗ +1 lies above a primary pivot. Proof . Let ( i , j ), . . . , ( i t ∗ , j t ∗ ) be the positions of primary pivots. A simple induction shows that C t ∗ +1 is theset of indices of columns of ˜∆ t ∗ +1 without primary pivots. The stopping criterium implies ˜∆ t ∗ +1 . C t ∗ +1 = 0. Finally,primary pivots, when marked, are, by the rules of the algorithm, the lowest nonzero entry of the column, and,by Proposition 14 (i), columns do not change after receiving a primary pivot mark. (cid:3) The next proposition establishes the equality between the final matrices produced by the Revised 1-blockIncremental Sweeping over F and the 1-block Incremental Sweeping Algorithm over F . There is of course nosense in looking for equality between other matrices in the sequence produced by the algorithm, since the orderof cancellation is in all likelihood quite different in the two algorithms. Proposition 16
Let ∆ ∈ F m × m be a connection matrix with column/row partition J , J . Let ˜∆ t ∗ +1 and ∆ m be the matrices obtained by applying the Revised 1-block Incremental Sweeping Algorithm over F and the 1-BlockIncremental Sweeping Algorithm over F to ∆ , respectively. Then ˜∆ t ∗ +1 = ∆ m and their primary pivots coincide. Proof . If ˜∆ t ∗ +1 = ∆ m , then their primary pivots coincide in position and value, by Corollaries 9 and 15.In both algorithms columns may suffer elementary column operations until they either reach zero or receivea primary pivot mark. Additionally, a column may suffer an elementary column operation only from anothercolumn with a primary pivot mark on its left and successive operations on a column may only increase the rangeof trailing zeros in that column, that is, the set of successive rows, ending in row m , containing zero entries. Theorder in which the primary pivots are identified probably differs between algorithms, but the important thingis that when an entry is eligible for receiving a primary pivot in either algorithm, the column it belongs to willhave been subjected to the same changes in both of them. In order for these changes to be the same in bothalgorithms, the active columns acting on them must be the same and the changes they provoke in a fixed columnmust occur in the correct order, from bottom up.The proof is by induction on the number of nonzero columns of ∆. If there is but one nonzero column, thenonzero entry at the bottom of this column will be marked by both algorithms as the unique primary pivot of∆, so t ∗ = 1, there will be no elementary column operations and in fact ˜∆ t ∗ +1 = ∆ m = ∆.Admit by induction that the last matrix of both algorithms coincide, when the number of nonzero columnsis smaller than k . Suppose ∆ has k nonzero columns. Let ( i , j ) be the first position to receive a primary pivotmark in the Revised 1-block Incremental Sweeping Algorithm over F . Then the entry ∆ j − i i j must also receivea primary pivot mark in the Incremental Sweeping Algorithm over F . To see that, note that, by definition of( i , j ), the entries in ∆ { i ,...,m }{ ,...,j − } and ∆ { i +1 ,...,m }{ ,...,m } are zero, and the number of trailing zeros can21nly increase. Consequently no entries in these submatrices may have been marked as a primary pivot before thesweeping of the j − i diagonal, so ∆ j − i i j is nonzero, with no primary pivot marks on its left or below it. So, inthis case, ∆ t ∗ +1 . j = ∆ m . j = ∆ . j . Furthermore, notice that, entries in positions ( i , j +1), . . . ( i , m ) will be sweptafter this and be marked, if nonzero, as change-of-basis entries in the Incremental Sweeping Algorithm over F ,since rows i +1, . . . , m of ∆ are zero and the entry in position ( i , j ) has a primary pivot. The changes possiblyeffected on columns due to these markings in the Incremental Sweeping Algorithm over F on the correspondingiterations are precisely the changes done in the first iteration of the Revised 1-block Incremental SweepingAlgorithm over F .Let ∆ (cid:48) ∈ F m × m be defined as follows:∆ (cid:48) . j = (cid:40) ˜∆ . j , if j (cid:54) = j , , otherwise.The matrix ∆ (cid:48) agrees with the matrix obtained from ∆ after the first iteration of the Revised 1-block IncrementalSweeping Algorithm over F , except for column j , which is zero. So ∆ (cid:48) encompasses the changes to columns dueto change-of-basis entries in row i , and has the j -th column equal to zero. Thus if we apply the IncrementalSweeping Algorithm over F to ∆ (cid:48) , the matrix ∆ (cid:48) m obtained agrees with ∆ m , except for column j , which wouldremain zero throughout the algorithm. Analogously, ∆ (cid:48) encompasses changes made to ∆ in the first iteration ofthe Revised 1-block Incremental Sweeping Algorithm over F , but differs from it in column j , which has beenzeroed. Thus the application of the Revised 1-block Incremental Sweeping Algorithm over F to ∆ (cid:48) produces amatrix ˜∆ (cid:48) t ∗ whose columns coincide with the corresponding ones from ˜∆ t ∗ +1 , except for the j -th column. Since∆ (cid:48) has at most k − (cid:48) m = ˜∆ (cid:48) t ∗ +1 . This implies∆ m . j = ∆ (cid:48) m . j = ˜∆ (cid:48) t ∗ . j = ˜∆ t ∗ +1 . j , for j (cid:54) = j . But since we already have that ∆ m . j = ˜∆ t ∗ +1 . j , we conclude that ∆ m = ˜∆ t ∗ +1 . (cid:3) A matrix is totally unimodular if all its square submatrices have determinant 0, 1 or −
1, see, for instance, [10].This property is invariant under transposition, multiplying a row or column by 0 , ±
1, adding or removing zerorows/columns or unit rows/columns. In this section we explore the additional facts that may be established whenthe input to the Spectral Sequence Sweeping Algorithm is restricted to totally unimodular connection matrices.An important class of connection matrices satisfying this property is that associated to Morse flows on surfaces.The set { , ± } is not closed under addition, so the appropriate algorithm to apply to totally unimodularconnection matrices is the Sweeping Algorithm over Z . Nevertheless, using the Block Sequential SweepingAlgorithm over F , the Revised 1-Block Incremental Sweeping Algorithm over F , Theorem 13 and the totalunimodularity property, we will show that the sequence of matrices and bases obtained when applying theIncremental Sweeping Algorithm over F to a TU connection matrix is compatible with the corresponding onesproduced by the application of Sweeping Algorithm over Z to ∆, in the following sense. The change from ∆ r to ∆ r +1 results from replacing basis element σ rj , of each column j containing a change-of-basis entry, with anintegral linear combination of elements of h with same chain index and associated with columns of index lessthan or equal j . The coefficients of this linear combination are calculated so as to zero out the entry in the22hange-of-basis position in ∆ r +1 , while maintaining the pattern of trailing zeros below it. Furthermore, amongstthe integral linear combinations that accomplish this, one must choose one with the smallest possible positiveleading coefficient. Even with this condition, there may be more than one optimal integral linear combination.The bases constructed by the Incremental Algorithm over F trivially satisfy the conditions pertaining the zeropatterns. We show that, for every r and j , the leading coefficient in the integral linear combination that is σ rj , produced by the application of the Incremental Algorithm over F to ∆, is 1, so it satisfies the optimalitycriterium, and the choice is thus compatible with the rules of the Sweeping Algorithm over Z .In the context of linear programming, the pivot operation on a nonzero entry a pq , called the pivot , of an m × n matrix A is defined as the following set of elementary row operations on A : 1) add to row i , for i (cid:54) = p , − a iq /a pq times row p , 2) divide row p by a pq . These operations may be expressed as the pre-multiplication of A by the m × m matrix B that coincides with the identity matrix except for its p -th column, which is given by b ip = − a iq a pq , if i (cid:54) = p, a pq , otherwise,see, for instance, [2]. The pivot operation assigns 1 to entry in position ( p, q ) and cancels entries on top andbelow position ( p, q ), so that column q of BA is equal to e p , the p -th canonical basis vector of R m . It follows fromLemma 9.2.2 of [15, p. 190], that, if A is totally unimodular, then BA also is. Now consider a variant of linearprogramming pivoting, where the m × m matrix (cid:101) B coincides with B except for entry in position ( p, p ), which isequal to 1. This means ( (cid:101) BA ) i . = ( BA ) i . , for i (cid:54) = p , and ( (cid:101) BA ) p . = a pq ( BA ) p . . If A is totally unimodular, then a pq = ±
1, so (cid:101) BA may be obtained from BA by at most a change in sign of row p of BA . But then, (cid:101) BA is alsototally unimodular. Proposition 17
Let ∆ ∈ { , ± } m × m be a totally unimodular connection matrix with column/row partition J , J . If we apply the Incremental Sweeping Algorithm over F to ∆ , then all primary pivots value either or − . Proof . First we analyze the application of the Revised 1-Block Sweeping Algorithm over F to ∆. Let ( i , j ),. . . , ( i t ∗ , j t ∗ ) be the positions of the primary pivots marked.We claim that ˜∆ t . C t is totally unimodular, for t = 1 , . . . , t ∗ + 1. This is trivially true for t = 1, by hypothesis.Assume it is true for t . Since( ˜∆ t T t ) . j = ˜∆ t . j , if j / ∈ C t , ˜∆ t . j , if j ∈ C t and j ≤ j t , ˜∆ t . j − ˜∆ ti t j ˜∆ ti t j t ˜∆ t . j t , if j ∈ C t and j > j t , the marking of an entry in position ( i t , j t ) of ˜∆ t . C t as a primary pivot implies the cancellation, in ˜∆ t +1 , of entrieson row i t , in columns in C t other than j t (although the update matrix construction provides the cancellationof entries to the right of column j t , entries to its left are zero, since ˜∆ ti t j t is the leftmost nonzero entry in row i t of ˜∆ t . C t ). Thus ˜∆ t +1 . C t = ( ˜∆ t T t ) . C t = ˜∆ t . C t T tC t C t . Notice that the cancellations are achieved by adding tocolumn j ∈ C t , j (cid:54) = j t , the appropriate multiple of column j t . But this is simply a transposed version of thevariant of the linear programming pivoting described above, with ˜∆ t . C t = A T and T tC t C t = (cid:101) B T . Therefore, if˜∆ t . C t is totally unimodular, then ˜∆ t +1 . C t is also totally unimodular. Since this property is, by definition, inheritedby submatrices, and C t +1 ⊂ C t , we conclude ˜∆ t +1 . C t +1 is also totally unimodular. By induction, ˜∆ t . C t is totallyunimodular for all t . 23his implies ˜∆ t . C t ∈ { , ± } m ×| C t | for all t . Hence all primary pivots marked in the application of the Revised1-Block Sweeping Algorithm over F to ∆ value 1 or − t ∗ +1 = ∆ m , the last matrix produced by the application of the Incremental SweepingAlgorithm over F to ∆. Therefore, by Corollaries 9 and 15, their primary pivots coincide in position and value. (cid:3) The elements are in place to establish that the primary pivots obtained when applying the IncrementalSweeping Algorithm over F to a TU connection matrix are unitary, see Theorem 18. This justifies its application,in place of the Sweeping Algorithm over Z , to TU connection matrices, see Proposition 19 and Corollary 20,even though the matrices produced during the sweeping may contain entries that would not have an inverse in Z . An example is shown in Figure 5. The highlighted entries in position (3 ,
4) and (7 ,
8) have been marked asprimary pivots during the sweeping of the first diagonal. No entry in the second diagonal is a primary pivot.When the third diagonal is swept, in ∆ , entries in (2 ,
5) and (6 ,
9) are marked as primary pivots, entry in (3 , , = 2. Since column 4 contains a primary pivot, row 4 will be zero atthe end of the algorithm. ∆ = h h h h h h h h h h h h h h h h h h h h (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) ∆ = Σ (cid:61) h Σ (cid:61) h (cid:43) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h (cid:43) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 5: ∆ obtained by Incremental Sweeping Algorithm over F applied to ∆ . Theorem 18 (Primary pivots for TU connection matrices)
Let ∆ be a TU connection matrix. Then theprimary pivots obtained when applying the Incremental Sweeping Algorithm over F thereto value ± . Proof . Suppose ∆ has column/row partition J , . . . , J b . The Uncoupling Theorem 13 justifies the applicationof the Block Sequential Sweeping Algorithm over F to ∆. If ∆ is TU, then ∆ J k − J k is TU, for k = 1, . . . , b ,and this property is maintained if one adds zero rows and/or columns to a matrix. Thus each ∆( k ), for k = 1,. . . , b , in the application of the Block Sequential Algorithm over F to ∆ is TU. Then the result follows fromProposition 17. (cid:3) roposition 19 Suppose that all the primary pivots obtained when applying the Incremental Sweeping Algorithmover F to an integral connection matrix ∆ are ± . Let σ r be the basis associated with the r -th matrix in thesequence produced by the algorithm, ∆ r . Then each basis element σ rj , for all r and j , is an integral linearcombination of the elements of h associated with columns to the left of, and including, column j , whose leadingcoefficient is . Proof . First of all, from the definition of T r in the Incremental Sweeping Algorithm over F and the hypothesesthat ∆ is integral and the primary pivots value ±
1, we obtain that T and ( T ) − , given by (10), are alsointegral. Then ∆ is integral and, applying induction, we may conclude that ∆ r is integral, for all r .Initially σ = h , so the property is true for r = 1. Suppose it is true for σ r . If ∆ r . j does not contain achange-of-basis, then σ r +1 j = σ rj and the result is true by the induction hypothesis. Suppose it does contain. It issufficient to consider one change-of-basis pivot, in a generic fixed position, say ( j − r, j ). Proposition 17 impliesthat the primary pivot to its left, say in position ( j − r, p ), is ±
1. Then, by the rules of the algorithm, σ r +1 j = σ rj − ∆ rj − r,j ∆ rj − r,p σ rp = σ rj ± ∆ rj − r,j σ rp = h jk + (cid:88) j (cid:48) Suppose that all the primary pivots obtained when applying the Incremental Sweeping Algorithmover F to an integral connection matrix ∆ are ± . Then the sequence of bases associated with the matrices ∆ , ∆ , . . . , is compatible with the application of the Sweeping Algorithm over Z to ∆ . In particular, thesequence produced by the application of the Incremental Sweeping Algorithm over F to a TU connection matrixis compatible with the application of the Sweeping Algorithm over Z thereto. Proof . The first result follows from Proposition 19. The second is a special case, since a surface connectionmatrix is integral and, by Theorem 18, the application of the Incremental Sweeping Algorithm over F theretoproduces unitary primary pivots. (cid:3) In the Revised 1-Block Incremental Sweeping Algorithm over F there are no change-of-basis pivots, each primarypivot “takes the lead” in the sense that all cancellations it would be responsible for in the Incremental SweepingAlgorithm over F are performed at once, as soon as it is marked. To achieve the same end matrix as theIncremental Sweeping Algorithm over F , the sweeping order is altered and only one primary pivot is marked periteration.This inspired the alteration of the Incremental Sweeping Algorithm over F to have all primary pivots havethis commandeering role. In the new algorithm we keep the same order of sweeping along the diagonals andthe criteria for marking an entry as a primary pivot. But the upper triangular unit-diagonal transition matrix25s calculated so that in the next matrix all entries to the right of the primary pivots are zeroed by means ofelementary column operations using exclusively the primary pivot columns. Although the computation of thetransition matrix is focused only on this post-multiplication aspect, it will be shown that the pre-multiplicationby its inverse does not interfere with the cancellations achieved in the post-multiplication.The new algorithm, called Row Cancellation Algorithm over F , is developed for generic connection matriceswith entries in F . Nevertheless, we will see that it can be applied to surface connection matrices, and, in thisspecial case, it is possible to give a topological/dynamic interpretation to its workings.The next proposition shows that the transition matrix with the properties described and with the desiredeffect upon the connection matrix is unique and gives a constructive formula therefor. Proposition 21 Let A ∈ F m × m and P = { j , . . . , j t } be a subset of column indices, such that j < · · · < j t and SP = { j − r, . . . , j k − r } ⊆ ..m , for some positive integral r . Suppose a j s − r,j s (cid:54) = 0 , while entries in column j s below a j s − r,j s are zero, for s = 1 , . . . , t . Then there exists a unique upper triangular unit-diagonal matrix T ∈ F m × m such that(i) the entries of T , strictly above the diagonal and outside rows with indices in P , are zero;(ii) ( AT ) j s − r,j s +1 ..m = 0 , for s = 1 , . . . , t .This matrix may be expressed as the product T · · · T t , where ( T s ) i . = e i , if i (cid:54) = j s ,e j s − a j s − r,j s (0 · · · a j s − r,j s +1 · · · a j s − r,m ) , ow. Proof . First we show that T = T · · · T t satisfies the conditions and then show its unicity. By construction, T s is a unit-diagonal upper triangular matrix satisfying (i), for s = 1, . . . , t . The post-multiplication of a matrix,say B , by T s has the effect of adding to column j of B a multiple of column j s of B , for j > j s . If B isunit-diagonal upper triangular and satisfies (i), these properties will be preserved by this operation. Thus, byinduction, T · · · T t is unit-diagonal, upper triangular and satisfies (i).By construction, columns 1 through j of A do not change with the post-multiplication by T . The entriesstrictly to the right of position ( j − r, j ) are zeroed out, since( AT ) j − r,j + k = a j − r,j · (cid:18) − a j − r,j a j − r,j + k (cid:19) + a j − r,j + k = 0 , for k = 1, . . . , m − j . Because entries of A below a j − r,j are zero by hypothesis, rows of A strictly below j − r do not change with the post-multiplication by T . The fact that j is the smallest element of P implies that( AT ) j s − r..m,j s = A j s − r..m,j s , for s = 1, . . . , t , and ( AT ) j − r +1 ..m,j = A j − r +1 ..m,j , for j > j .Assume by induction that ( AT · · · T s ) j − r..m,j = A j − r..m,j , for j = j , . . . , j t ,( AT · · · T s ) j s − r +1 ..m,j = A j s − r +1 ..m,j , for j > j s , and that entries of AT · · · T s strictly to the right of positions( j − r, j ), . . . , ( j s − r, j s ) are zero. The post-multiplication of AT · · · T s by T s +1 has the effect of addingto column ( AT · · · T s ) . j a multiple of column ( AT · · · T s ) . j s +1 , for j > j s +1 . Thus columns 1 through j s +1 of ( AT · · · T s ) do not change with the post-multiplication by T s +1 and, in particular, ( AT · · · T s +1 ) j − r..m,j =26 j − r..m,j , for j = j , . . . , j s +1 . If j > j s +1 ,( AT · · · T s T s +1 ) . j = ( AT · · · T s ) T s +1 . j = ( AT · · · T s ) . j − a j s +1 − r,j s +1 a j s +1 − r,j ( AT · · · T s ) . j s +1 . Then, for j > j s +1 , the induction hypothesis implies( AT · · · T s T s +1 ) j k − r,j = − a j s +1 − r,j s +1 a j s +1 − r,j · , if k = 1 , . . . , s,a j s +1 − r,j − a j s +1 − r,j s +1 a j s +1 − r,j · a j s +1 − r,j s +1 = 0 , if k = s + 1 . So the entries to the right of ( j − r, j ), . . . , ( j s +1 − r, j s +1 ) are zero in AT · · · T s +1 .The fact that ( AT · · · T s ) j s +1 − r +1 ..m,j s +1 = A j s +1 − r +1 ..m,j s +1 = 0 implies that ( AT · · · T s +1 ) j s +1 − r +1 ..m,j =( AT · · · T s ) j s +1 − r +1 ..m,j = A j s +1 − r +1 ..m,j , for j > j s +1 . In particular, ( AT · · · T s +1 ) j − r..m,j = A j − r..m,j , for j = j s +2 , . . . , j t , since j < · · · < j t . Applying induction, we obtain (ii).To show unicity, we translate into equations the conditions T must satisfy. Pick j ∈ { , . . . , m } such that j s < j ≤ j s +1 . Column j of T has entries already determined from its general description: t jj = 1 and t ij = 0,for i / ∈ { j , j , . . . , j s , j } . The set of conditions in (ii) imply the remaining entries must satisfy the linear system A SP s P s T P s ,j + A SP s ,j = 0 , where SP s = { j − r, . . . , j s − r } and P s = { j , . . . , j s } . By hypothesis, A SP s P s is upper triangular with nonzerodiagonal entries. Thus the linear system above has a unique solution, which must coincide with the correspondingentries of T · · · T t . (cid:3) The Row Cancellation Algorithm over F is presented below. Its properties will be the subject of the followingproposition. Notice that, unlike the Incremental Sweeping Algorithm over F , this one does not need a final step.This follows from the fact that the only marks assigned during the sweeping of a diagonal are primary pivots.Even if a primary pivot mark is assigned to the only entry in diagonal m − 1, it will not provoke any changes tothe connection matrix, since there are no entries to its right to cancel. Consequently, no transition matrix needbe computed if r = m − Row Cancellation Algorithm over F Input: nilpotent upper triangular matrix ∆ ∈ F m × m with column/row partition J , J , . . . , J b . Initialization Step: r = 0 (cid:101) ∆ r = ∆ (cid:101) T r = I ( m × m identity matrix) Iterative Step: (Repeated until all diagonals parallel and to the right of the main diagonal have been swept) Matrix ∆ update r ← r + 1 (cid:101) ∆ r = ( (cid:101) T r − ) − (cid:101) ∆ r − (cid:101) T r − Markup Sweep entries of (cid:101) ∆ r in the r -th diagonal: If (cid:101) ∆ rj − r,j (cid:54) = 0 and (cid:101) ∆ r . ,j does not contain a primary pivot Then permanently mark (cid:101) ∆ rj − r,j as a primary pivot Matrix (cid:101) T r construction If diagonal r has no primary pivots or r = m − Then (cid:101) T r = I Else Let j < · · · < j t r be the indices of columns with primary pivots indiagonal r (cid:101) T r = (cid:101) T r, · · · (cid:101) T r,t r , where( (cid:101) T r,s ) i . = e i , if i (cid:54) = j s ,e j s − (cid:101) ∆ rj s − r,j s (0 · · · (cid:101) ∆ rj s − r,j s +1 · · · (cid:101) ∆ rj s − r,m ) , ow. Output: ( (cid:101) ∆ , . . . , (cid:101) ∆ m − ) and ( (cid:101) T , . . . , (cid:101) T m − )To keep up the parallel with the Incremental Sweeping Algorithm over F , we denote by (cid:101) P r the set of columnindices of primary pivots on or below diagonal r in (cid:101) ∆ m − . Notice that the off-diagonal nonzero entries of (cid:101) T r belong to rows with indices equal to the column indices of the primary pivots in diagonal r . Using induction, it isstraightforward that the off-diagonal nonzero entries in (cid:101) T · · · (cid:101) T r belong to rows in (cid:101) P r . When (cid:101) ∆ r +1 is obtainedin the update step, we will see in Proposition 22 that the entries to the right of the primary pivots on the r -thdiagonal are zeroed out. Here the parallel breaks, because the changes from ∆ r to ∆ r +1 in the IncrementalSweeping Algorithm over F arise from the cancellation of change-of-basis entries on the r -th diagonal, usingprimary pivots identified in previous iterations, and thus belonging to diagonals 1, . . . , r − 1. Thus the off-diagonal nonzero entries of T r belong to P r − \P r − and, by induction, the off-diagonal nonzero entries of T · · · T r belong to P r − . Finally, the condition that the candidate for primary pivot do not have a primarypivot to its left is not required in this algorithm, due to Proposition 22 (vii).The following proposition mimics the results obtained for the Sweeping Algorithm over Z and the IncrementalSweeping Algorithm over F . Proposition 22 Let ( (cid:101) ∆ (= ∆) , (cid:101) ∆ , . . . , (cid:101) ∆ m − ) and ( (cid:101) T = I, (cid:101) T , . . . , (cid:101) T m − ) be the sequences of connection andtransition matrices produced by the application of the Row Cancellation Algorithm over F to the connectionmatrix ∆ ∈ F m × m with column/row partition J , . . . , J b . Then, for r = 0 , , . . . , m − , we have:(i) (cid:101) ∆ r is compliant with the allowable sparsity pattern of ∆ ;(ii) the nonzero entries of (cid:101) ∆ r strictly below the r -th diagonal are either primary pivots (always nonzero) or lieabove a unique primary pivot;(iii) if (cid:101) ∆ sp − r,p is a primary pivot, then (cid:101) ∆ rp . does not contain a primary pivot, for s ≤ r ;(iv) if (cid:101) ∆ rp − r,p is marked as a primary pivot, then (cid:101) ∆ r +1 p − r,p +1 ..m = 0 ;(v) if (cid:101) ∆ rp − r,p is marked as a primary pivot, then (cid:101) ∆ sp . = 0 , for s ≥ r + 1 .Additionallly, vi) If (cid:101) ∆ m − ij is a primary pivot, then (cid:101) ∆ . i has no primary pivot;(vii) if (cid:101) ∆ rp − r,p is marked as a primary pivot, then (cid:101) ∆ sp − r,p +1 ..m = 0 , for s ≥ r + 1 ;(viii) each row of (cid:101) ∆ m − may contain at most one primary pivot. Proof . Assertion (i) is true for r = 0. Suppose it is true for arbitrary fixed r . If there are no primary pivots onthe r -th diagonal, it trivially holds for r + 1, since (cid:101) ∆ r +1 = (cid:101) ∆ r in this case.Now suppose (1 < ) j < · · · < j t r are the column indices of primary pivots in diagonal r . Then the transitionmatrix (cid:101) T r is the product (cid:101) T r = (cid:101) T r, · · · (cid:101) T r,t r , where( (cid:101) T r,s ) i . = e i , if i (cid:54) = j s ,e j s − (cid:101) ∆ rj s − r,j s (0 · · · (cid:101) ∆ rj s − r,j s +1 · · · (cid:101) ∆ rj s − r,m ) , ow. (14)The inverse of (cid:101) T r,s is given by( (cid:101) T r,s ) − i . = e i , if i (cid:54) = j s ,e j s + 1 (cid:101) ∆ rj s − r,j s (0 · · · (cid:101) ∆ rj s − r,j s +1 · · · (cid:101) ∆ rj s − r,m ) , ow. (15)The connection matrix update may be computed in the following nested fashion:1st step: (cid:101) ∆ r (cid:101) T r, = (cid:101) ∆ r, (cid:101) T r, = (cid:101) ∆ r, (cid:101) T r, ) − (cid:101) ∆ r, = (cid:101) ∆ r, (cid:101) ∆ r, (cid:101) T r, = (cid:101) ∆ r, (cid:101) T r, ) − (cid:101) ∆ r, = (cid:101) ∆ r, ...By induction, the matrix (cid:101) ∆ r = (cid:101) ∆ r, satisfies (i). We will use induction in the nested product to prove thatthe last matrix (cid:101) ∆ r, t r = (cid:101) ∆ r +1 also does. Suppose (cid:101) ∆ r, s − satisfies (i). In the post-multiplication by (cid:101) T r,s , tocolumn j s + j of (cid:101) ∆ r, s − we add − (cid:101) ∆ rj s − r,j s + j / (cid:101) ∆ rj s − r,j s times column j s , for j = 1 , . . . , m − j s . This coefficient isnonzero only if columns j s and j s + j belong to the same subset of the column partition. Since column j s is tothe left of column j s + j , and (cid:101) ∆ r, s − is compliant with the sparsity pattern of ∆, this property is inherited bythe product (cid:101) ∆ r, s − (cid:101) T r,s , so (cid:101) ∆ r, s − satisfies (i). Using (15), in the pre-multiplication by ( (cid:101) T r,s ) − , to row j s of (cid:101) ∆ r, s − we add (cid:101) ∆ rj s − r,j s + j / (cid:101) ∆ rj s − r,j s times row j s + j , for j = 1 , . . . , m − j s . Again the multiplication coefficientis nonzero only if the two rows belong to the same subset of the row partition. Given that row j s lies above row j s + j and that (cid:101) ∆ r, s − is compliant with ∆’s allowable sparsity pattern, it follows that (cid:101) ∆ r, s = ( (cid:101) T r,s ) − (cid:101) ∆ r, s − is compliant with the allowable sparsity pattern of ∆. By induction, we conclude that (cid:101) ∆ r +1 satisfies (i). Byinduction, (i) is satisfied for all r .Let f be the first nonzero diagonal of ∆ and (1 < ) j < · · · < j t f be the columns of the nonzero entries. Then∆ = (cid:101) ∆ = · · · = (cid:101) ∆ f . Given the construction of (cid:101) T r and Proposition 21, we have that ( (cid:101) ∆ r (cid:101) T r ) j − r,j +1 ..m = 0, for j ∈ { j , . . . , j t r } .Since there are no primary pivots to the left of diagonal f , (iii) is trivially valid for s < f . The pre-multiplication of (cid:101) ∆ f (cid:101) T f by ( (cid:101) T r ) − may be obtained by the sequence of pre-multiplications: by ( (cid:101) T r, ) − , by( (cid:101) T f, ) − , etc. Therefore, given the structure of ( (cid:101) T f,j ) − in (15), only rows j , . . . , j t f , are affected. Supposethere exists j i ∈ { j , . . . , j t f } such that (cid:101) ∆ fj i . contains a primary pivot. Since entries to the left and entries belowa primary pivot in (cid:101) ∆ f are zero, and primary pivots are nonzero, we have (cid:101) ∆ fj i − f . (cid:101) ∆ f . j i + f = (cid:101) ∆ fj i − f,j i (cid:101) ∆ fj i ,j i + f (cid:54) = 0 , (cid:101) ∆ f . Therefore (iii) is also true for s = f .Given the construction of (cid:101) T f and Proposition 21, we have that( (cid:101) ∆ f (cid:101) T f ) j − f,j = (cid:101) ∆ fj − f,j (cid:54) = 0 , for j ∈ { j , . . . , j t f } and ( (cid:101) ∆ f (cid:101) T f ) j − f,j +1 ..m = 0 , for j ∈ { j , . . . , j t f } . Recall that the pre-multiplication by ( (cid:101) T f ) − only affects rows j , . . . , j t f of (cid:101) ∆ f (cid:101) T f , none of which contains aprimary pivot. This implies that (cid:101) ∆ f +1 j − f,j = (( (cid:101) T f ) − (cid:101) ∆ f (cid:101) T f ) j − f,j = (cid:101) ∆ fj − f,j (cid:54) = 0 , for j ∈ { j , . . . , j t f } (16)and (( (cid:101) T f ) − (cid:101) ∆ f (cid:101) T f ) j − f,j +1 ..m = ( (cid:101) ∆ f (cid:101) T f ) j − f,j +1 ..m = 0 , for j ∈ { j , . . . , j t f } . (17)Hence (iv) is valid for r = f .The fact that entries to the left of primary pivots are zero, (17) and the nilpotency of (cid:101) ∆ f +1 imply that0 = (cid:101) ∆ f +1 j − f . (cid:101) ∆ f +1 = (cid:101) ∆ f +1 j − f,j (cid:101) ∆ f +1 j . , for j ∈ { j , . . . , j t f } . (18)Using (16) and (18), we have that (cid:101) ∆ f +1 j . = 0 , for j ∈ { j , . . . , j t f } , (19)which partly proves (v), for r = f , when s = f + 1.The post-multiplication part of the update, in the transition from (cid:101) ∆ f to (cid:101) ∆ f +1 , may only change entriesstrictly to the right of diagonal f , since entries below a primary pivot are zero. From (19), we see that thepre-multiplication part of the update affects exclusively rows j , . . . , j t f , which are zeroed, and so does notcreate new nonzero entries on or below the f -th diagonal. Furthermore, since rows j , . . . , j t f do not containprimary pivots by (iii), the primary pivots remain nonzero in (cid:101) ∆ f +1 . We conclude that (ii) true for r = f + 1.Elementary column operations do not change a zero row, so to establish the rest of (v) we must show thatthe elementary row operations done in the pre-multiplication part of future updates do not change rows j , . . . , j t f . Rows that change in an update are precisely those whose indices coincide with the columns that receiveprimary pivot marks in that iteration. But a column may contain at most one primary pivot. Therefore rows j , . . . , j t f will remain invariant for the rest of the execution of the algorithm and we conclude that (v) is truefor r = f .Suppose (ii)–(v) are true for (cid:101) ∆ , . . . , (cid:101) ∆ r . If all the nonzero entries of the r -th diagonal of (cid:101) ∆ r lie aboveprimary pivots, then this diagonal receives no primary pivot marks, (cid:101) ∆ r +1 = (cid:101) ∆ r , and (ii)–(v) are trivially truefor r + 1.Now consider the remaining case, where there are nonzero entries on the r -th diagonal of (cid:101) ∆ r with no primarypivots below. These will be marked as primary pivots and (ii) implies all entries below these primary pivots ofthe r -th diagonal in (cid:101) ∆ r are zero. Let j < · · · < j t r be the indices of the columns containing these entries. Then,using the nilpotency of (cid:101) ∆ r and (ii), we have0 = (cid:101) ∆ rj − r . (cid:101) ∆ r . j + r = m (cid:88) k =1 (cid:101) ∆ rj − r,k (cid:101) ∆ rk,j + r = j (cid:88) k =1 (cid:101) ∆ rj − r,k (cid:101) ∆ rk,j + r , for j ∈ { j , . . . , j t r } . (cid:101) ∆ rj − r,j lie strictly below the r -th diagonal. Using the induction hypothesis (ii),these nonzero entries lie above primary pivots. But then, using (v), we obtain0 = (cid:88) k ∈ (cid:101) P r − ∩{ i | i 1, was swept, then, by (v), (cid:101) ∆ j − ii . i = 0,contradicting the fact that (cid:101) ∆ j − iij (cid:54) = 0. Thus (vi) is true.If (cid:101) ∆ j − iij is marked as a primary pivot, then the algorithm’s rules implies column j will not be marked againin future iterations. Item (vii) is true for s = r + 1 by item (iv). The validity of (vii) for larger values of s followsfrom the facts that (a) elementary column operations involving columns j + 1, . . . , m will not change this trailingzero pattern, so primary pivots marked in these columns in future iterations do not alter this part of the row,(b) primary pivots marked in future iterations, so in diagonals to the right of diagonal j − i , in columns in therange 1 ..j − i , and, by (ii), the entries below these primary pivots are zero,so elementary column operations caused by these markings will not interfere with the zeros to the right of theprimary pivot in position ( i, j ), and (c) by (vi) column i will not receive a primary pivot mark, so row i will notbe altered by elementary row operations.Finally, (viii) is a consequence of (vii) and the rules of the algorithm, since an entry must be nonzero to beeligible for receiving a primary pivot mark. (cid:3) This time the complementary relationship between a column p containing a primary pivot and row p in thelast matrix in the sequence produced by the Row Cancellation Algorithm over F is a straightforward consequenceof Proposition 22. Corollary 23 Let (cid:101) ∆ m − be the last connection matrix in the sequence produced by the application of the RowCancellation Algorithm over F to the connection matrix ∆ ∈ F m × m with column/row partition J , . . . , J b . Then (cid:101) ∆ m − . j (cid:101) ∆ m − j . = 0 , for all j. (23) Proof . By Proposition 22 (ii) for r = m − 1, the only nonzero columns of (cid:101) ∆ m − are those containing primarypivots. But if column p contains a primary pivot, then by Proposition 22 (v), (cid:101) ∆ m − p . = 0. Thus, in either case,(23) is verified. (cid:3) Another feature that is shared with the Incremental Sweeping Algorithm over F is the possibility of doingthe matrix update in blocks. Lemma 24 The matrix updates in the application of the Row Cancellation Algorithm over F to the connectionmatrix ∆ ∈ F m × m with column/row partition J , . . . , J b can be done in a blockwise fashion as follows. (cid:101) ∆ rJ k − J k = ( (cid:101) T r − J k − J k − ) − (cid:101) ∆ r − J k − J k (cid:101) T r − J k J k , for k = 1 , . . . , b. (24) Proof . The result is nontrivial only if (cid:101) T r − is not equal to the identity matrix. Suppose therefore that columns j , . . . , j t r contain primary pivot marks at diagonal r − A and B are two upper triangular, unit diagonal matrices in F m × m such that the positions of their nonzeroentries is contained in ∪ bk =0 J k × J k , then AB is trivially upper triangular with unit diagonal. The entry of AB in position ( i, j ) is given by ( AB ) ij = m (cid:88) s =1 a is b sj . The sum is zero unless there is some s and k such that i , j , and s belong to J k . But this is possible only if i and j belong to J k . Thus the support of AB is also contained in ∪ bk =0 J k × J k .The expression of (cid:101) T r − ,s in (14) implies (cid:101) T r − ,si,j (cid:54) = 0 only if i = j or i = j s and j belongs to the subset of thepartition that contains j s . Thus the support of (cid:101) T r − ,s is contained in ∪ bk =0 J k × J k . Using the argument in theprevious paragraph and induction, we conclude that (cid:101) T r − is upper triangular, has unit diagonal and its supportis contained in ∪ bk =0 J k × J k . Given the structure of ( (cid:101) T r − ,s ) − in (15), the same is true about ( (cid:101) T r − ) − . Thisimplies (( (cid:101) T r − ) − (cid:101) ∆ r − (cid:101) T r − ) J k − J k = ( (cid:101) T r − ) − J k − . (cid:101) ∆ r − (cid:101) T r − . J k = ( (cid:101) T r − ) − J k − J k − (cid:101) ∆ r − J k − J k (cid:101) T r − J k J k . (25)Lemma 3 implies ( (cid:101) T r − ) − J k − J k − = ( (cid:101) T r − J k − J k − ) − . This and (25) conclude the proof. (cid:3) The last ingredient to prove that the Row Cancellation Algorithm over F can be done sequentially is theanalogue of Lemma 12 below. It concerns the application of the Row Cancellation Algorithm over F to aconnection matrix ∆ ∈ F m × m , with column/row partition J , . . . , J b . We let (cid:101) J k be the set of columns in J k that contain primary pivot entries in (cid:101) ∆ m − , for k = 1 , . . . b . Additionally, (cid:101) J k = J k \ (cid:101) J k , for k = 1 , . . . b . Lemma 25 Suppose the Row Cancellation Algorithm over F is applied to the connection matrix ∆ ∈ F m × m withcolumn/row partition J , . . . , J b . Then the markings during the sweeping of the r -th diagonal of (cid:101) ∆ r on entriesin columns belonging to J k and the construction of (cid:101) T rJ k J k are completely determined by the values of the entriesin (cid:101) ∆ r (cid:101) J k − J k . Proof . Let j ∈ J k . Since rows in (cid:101) J k − do not receive primary pivot marks, we may assume that j − r ∈ (cid:101) J k − .If (cid:101) ∆ j − r,j = 0, it will not be marked. If (cid:101) ∆ rj − r,j (cid:54) = 0, one must check whether there is a primary pivot below it.It there is, it must belong to a row in (cid:101) J k − . If there is not, then this entry will be marked as a primary pivotand contribute to one of the matrices, say (cid:101) T r,s whose product constitute (cid:101) T r is the identity with the additionof the vector − / (cid:101) ∆ rj − r,j (0 · · · (cid:101) ∆ rj − r,j +1 · · · (cid:101) ∆ j − r,m ) to the j -th row. The entry − (cid:101) ∆ rj − r,j + t / (cid:101) ∆ rj − r,j in this vectorare nonnull only if j + t also belongs to J k . Since we already have that j − r ∈ (cid:101) J k − , the result is proved. (cid:3) The validity of the block update established Lemma 25 and the complementarity relationship between acolumn with a primary pivot and the row of same index given in Corollary 23 give rise to a simplified versionof the Row Cancellation Algorithm over F . The Block Sequential Row Cancellation Algorithm over F is astraightforward adaptation of the Block Sequential Sweeping Algorithm over F , where, at step k , instead ofapplying the Incremental Sweeping Algorithm over F to ∆( k ), one applies the Row Cancellation Algorithm over F thereto. The proof of the corresponding Uncoupling Theorem is a straightforward adaptation of the originalone. 33 heorem 26 (Row Cancellation Uncoupling) Let ∆ ∈ F m × m be a connection matrix with row/columnpartition J , . . . , J b . Let (cid:101) ∆ m − be the matrix produced by the application of the Row Cancellation Algorithmover F to ∆ , and let (cid:101) ∆( k ) m − , for k = 1 , . . . , b , be the matrices obtained in the Block Sequential Row CancellationAlgorithm over F applied to ∆ . Then (cid:101) ∆ m − J k − J k = (cid:101) ∆( k ) m − J k − J k , for k = 1 , . . . , b and the collection of primary pivotsencountered in the application of the Row Cancellation Algorithm over F to ∆( k ) , for k = 1 , . . . , b , coincideswith the primary pivots found when it is applied to ∆ . Theorem 26 significantly simplifies the next results, since it allows us to consider connection matrices with onlyone block, which means only elementary column operations need be performed in the matrix update step. Thisspecial instance of the Row Cancellation Algorithm over F will be called 1-Block Row Cancellation Algorithmover F . Notice that, although the proposition guarantees the equalities of the primary pivots up to r = m − m − and of ∆ m are equal. Proposition 27 Let ∆ ∈ F m × m be a connection matrix with row/column partition J , . . . , J b . Let ∆ , . . . , ∆ m and T , . . . , T m − (resp., (cid:101) ∆ , . . . , (cid:101) ∆ m − and (cid:101) T , . . . , (cid:101) T m − ) be the matrices produced in the application ofthe Incremental Sweeping Algorithm over F (resp., Row Cancellation Algorithm over F ) to ∆ . Then the primarypivots of ∆ r and (cid:101) ∆ r coincide in position and value, for r = 1 , . . . , m − . Proof . By Theorem 26, it is enough to prove the result for a one block connection matrix. In this case, onlycolumn operations are performed in either algorithm.The position and value equalities are trivially true for primary pivots on the first diagonal. Assume theequalities hold for those in diagonals 1, . . . , r − 1. This implies, in particular, that P r − = (cid:101) P r − . We will proveboth equalities for a generic pivot on diagonal r .Suppose ∆ rj ∗ − r,j ∗ is marked as primary pivot in the application of the 1-Block Incremental Sweeping Algo-rithm over F to ∆. This means that there are no primary pivots on its left or below it in ∆ r . Since these entrieslie below the r -th diagonal, the same is true for (cid:101) ∆ r . Furthermore, Proposition 8 (ii) and Proposition 22 (ii)imply that ∆ rj ∗ − r +1 ..m,j ∗ = 0 , (26) (cid:101) ∆ rj ∗ − r +1 ..m,j ∗ = 0 . (27)The update rules of the 1-Block Incremental Sweeping Algorithm over F , imply that each column, say j , ofeach matrix in the sequence ∆ , ∆ , . . . produced by the algorithm, is the sum of the original column plus alinear combination of primary pivot columns, i.e., columns that have already received a primary pivot mark, onits left. Each addition of a multiple of a primary pivot column aims to increase the number of trailing zeros incolumn j . Therefore, if i max = max { i | ∆ rij (cid:54) = 0 } , then ∆ (cid:96) . j may be expressed as∆ (cid:96) . j = ∆ . j + α ˙∆ r . j + · · · + α t ˙∆ r t . j t , (28)where (the dot indicates that) ˙∆ r i is the first matrix in the sequence that contains a primary pivot mark incolumn j i , j i < j , r i < (cid:96) and the primary pivot in column j i lies on a row strictly below i max , for i = 1 , . . . , t .Doing this recursively, after a finite number of steps we reach an expression involving only original columns,transforming (28) into ∆ (cid:96) . j = ∆ . j + ∆ . P (cid:96),j β (cid:96),j , (29)34here P (cid:96),j ( ⊂ P (cid:96) − ) is the set of column indices with primary pivot entries in the submatrix ∆ (cid:96)i max +1 ..m, ..j − ,that is, with entries in rows strictly below row i max and in columns strictly to the left of column j , and β (cid:96),j is acolumn vector of appropriate dimension.When (cid:96) = r and j = j ∗ , (29) becomes ∆ r . j ∗ = ∆ . j ∗ + ∆ . P r,j ∗ β r,j ∗ . (30)Applying this procedure to each column of ∆ r in P r,j ∗ we obtain the following matrix equality∆ r . P r,j ∗ = ∆ . P r,j ∗ B r, P r,j ∗ , (31)where B r, P r,j ∗ is a square matrix. Submatrix ∆ r . P r,j ∗ has full column rank, since, by a column permutation,it can be cast in a column echelon format. Thus B r, P r,j ∗ is invertible and ∆ . P r,j ∗ also has full column rank.Additionally, since the primary pivots in ∆ r . P r,j ∗ lie strictly below row j ∗ − r , we have have that∆ rj ∗ − r +1 ..m, P r,j ∗ = ∆ j ∗ − r +1 ..m, P r,j ∗ B r, P r,j ∗ , (32)and both ∆ rj ∗ − r +1 ..m, P r,j ∗ and ∆ j ∗ − r +1 ..m, P r,j ∗ have full column rank.It is easy to see that each column, say j , in each matrix of the sequence generated by the application ofthe 1-Block Row Cancellation Algorithm over F to ∆, say (cid:101) ∆ (cid:96) , is the sum of the original column plus a linearcombination of primary pivot columns to its left. But in this algorithm, this linear combination may contain aprimary pivot column whose primary pivot entry lies above the lowest nonzero entry of (cid:101) ∆ (cid:96) . j .Since, by induction the primary pivot entries below diagonal r in ∆ r and (cid:101) ∆ r coincide in position and value,the column indices of primary pivot entries in (cid:101) ∆ rj ∗ − r +1 ..m, ..j ∗ − is precisely P r,j ∗ . Then (cid:101) ∆ r . j ∗ = ∆ . j ∗ + (cid:88) j ∈P r,j ∗ α j ˙ (cid:101) ∆ r j . j + (cid:88) j ∈ P r − \P r,j ∗ j < j ∗ α j ˙ (cid:101) ∆ r j . j , (33)where, in both sums, ˙ (cid:101) ∆ r j denotes the first matrix in the sequence that contains a primary pivot in column j . Thesecond summation in the right-hand-side of (33) aggregates the contribution to (cid:101) ∆ r . j ∗ of primary pivot columns in P r − , to the left of column j ∗ , and whose primary pivots lie strictly above j ∗ − r (remember that, by induction,there are no primary pivots to the left of (cid:101) ∆ rj ∗ − r,j ∗ ).Express each ˙ (cid:101) ∆ r j . j in the first summation in an analogous way, aggregating in a second and separate sum thecontribution of primary pivot columns with primary pivot entries strictly above row j ∗ − r . Insert back into itsrespective place in (33) the expression for each ˙ (cid:101) ∆ r j . j , for j ∈ P r,j ∗ , to obtain (cid:101) ∆ r . j ∗ = ∆ . j ∗ + (cid:88) j ∈P r,j ∗ α j ∆ . j + (cid:88) j ∈P r,j ∗ α (cid:48) j ˙ (cid:101) ∆ r j . j + (cid:88) j ∈ P r − \P r,j ∗ j < j ∗ α (cid:48) j ˙ (cid:101) ∆ r j . j , (34)where the primes in the second summation indicate that the contribution of the ‘second sum’ of the expressionof each ˙ (cid:101) ∆ r j . j has already been incorporated therein. Furthermore, although this is not explicitly shown in (34),max { r j | α (cid:48) j (cid:54) = 0 , j ∈ P r,j ∗ } < max { r j | α j (cid:54) = 0 , j ∈ P r,j ∗ } . This follows from the fact that, when one expressesa column, say (cid:101) ∆ r j . j , as a sum of ∆ . j and a linear combination of primary pivot columns, these primary pivotcolumns were identified in iterations previous to the sweeping of the r j -th diagonal and so are associated with35ower numbered diagonals. If we repeat this procedure a sufficient (and finite) number of times, eventually allcoefficients in the middle sum in the right-hand-side of (34) will be zero, and we will have (cid:101) ∆ r . j ∗ = ∆ . j ∗ + ∆ . P r,j ∗ (cid:101) β r,j ∗ + (cid:88) j ∈ P r − \P r,j ∗ j < j ∗ γ j ˙ (cid:101) ∆ r j . j . (35)Notice that ˙ (cid:101) ∆ r j j ∗ − r,j = 0 , for j ∈ P r − \P r,j ∗ , j < j ∗ , (36)since the primary pivot entries of the columns in P r − \P r,j ∗ , to the left of j ∗ , are strictly above row j ∗ − r andentries below a primary pivot are zero, see Proposition 22 (ii).Using (26), (27), (30), (35) and (36), we have0 = ∆ j ∗ − r +1 ..m,j ∗ + ∆ j ∗ − r +1 ..m P r,j ∗ β r,j ∗ , (37)0 = ∆ j ∗ − r +1 ..m,j ∗ + ∆ j ∗ − r +1 ..m P r,j ∗ (cid:101) β r,j ∗ . (38)Since ∆ j ∗ − r +1 ..m P r,j ∗ has full column rank, the solution to the (feasible) linear systems (37) and (38) isunique, so β r,j ∗ = (cid:101) β r,j ∗ . (39)But then, (30), (35), (36) and (39) imply that∆ rj ∗ − r,j ∗ = ∆ j ∗ − r,j ∗ + ∆ j ∗ − r, P r,j ∗ β r,j ∗ = ∆ j ∗ − r,j ∗ + ∆ j ∗ − r, P r,j ∗ (cid:101) β r,j ∗ = (cid:101) ∆ rj ∗ − r,j ∗ . We’ve proved both entries are equal in value. By induction hypothesis, there are no primary pivots to its left orbelow it in the corresponding matrices. Thus (cid:101) ∆ rj ∗ − r,j ∗ = ∆ rj ∗ − r,j ∗ (cid:54) = 0 will also be marked as a primary pivot.The proof for the converse is analogous. Thus, by induction, the result follows. (cid:3) Theorem 18, Corollary 20 and Proposition 27 imply that the primary pivots of (cid:101) ∆ m − , the last matrixproduced by the application of the Row Cancellation Algorithm over F to the order m TU connection matrix ∆,are either 1 or − 1. This implies the transition matrices (and their inverses) produced by the application of theRow Cancellation Algorithm over F to a TU connection matrix are all integral. This justifies the application ofthe Row Cancellation Algorithm over F to TU connection matrices. Hence, we have the following result. Theorem 28 (Equality of primary pivots) Let ∆ ∈ {− , , } m be a TU connection matrix with column/rowpartition J , . . . , J b . The primary pivots marked in the application of the Sweeping Algorithm over Z to ∆ co-incide in value and position with the primary pivots marked in the application of Row Cancellation Algorithmover F thereto. Notice that, although the primary pivots of ∆ m and (cid:101) ∆ m − coincide, the matrices might differ, as exemplifiedin Figure 6. In this figure, cells containing primary entries are highlighted.36urface connection matrix ∆ = h h h h h h h h h h h h h h h h h h h h (cid:45) (cid:45) (cid:45) (cid:45) Incremental Sweeping Algorithm over F Smale’s Cancellation Algorithm over F Σ (cid:61) h (cid:43) h (cid:43) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ (cid:45) (cid:45) (cid:45) (cid:45) Σ (cid:61) h (cid:43) h (cid:43) h Σ (cid:61) h (cid:43) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ (cid:61) h Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ (cid:45) (cid:45) (cid:45) (cid:45) Figure 6: Surface connection matrix and the last matrices produced by the application thereto of the IncrementalSweeping Algorithm over F and Smale’s Cancellation Algorithm over F .37 Dynamical interpretation for the algebraic cancellations In this section we will prove Theorem 1. Proof . (Theorem 1 ) Recall that M is a closed simply connected manifold of dimension m > f : M → R a Morse-Smale function. Also ( C, ∆) is a filtered Morse chain complex with finest filtration and ( E r , d r ) is theassociated spectral sequence. Assume that ∆ is a totally unimodular matrix.As proven in [4], the non zero differentials of the spectral sequence are induced by the pivots. When workingwith TU connection matrices, it follows from Theorem 18 (Primary Pivots for TU Connection Matrices) thatthe primary pivots are always equal to ± 1. Hence, the differentials d rp : E rp → E rp − r associated to primary pivotsare isomorphisms and the ones associated to change of basis pivots always correspond to zero maps. In fact, ifa differential d rp : E rp → E rp − r corresponds to a change of basis pivot, then there is a primary pivot in row p − r and thus E rp − r = 0. Consequently, the non zero differentials are isomorphisms and this implies that at the nextstage of the spectral sequence they produce algebraic cancellations , i.e. E r +1 p = E r +1 p − r = 0.The Equality of Primary Pivots Theorem (Theorem 28) proves that primary pivots marked when applyingthe Sweeping Algorithm over Z to ∆ coincide in value and position with primary pivots marked when applyingthe Row Cancellation Algorithm over F thereto. Therefore, the algebraic cancellations of the modules E r of thespectral sequence determined by the SSSA are in one-to-one correspondence with the primary pivots determinedby the RCA.In what follows, we analyze the dynamics within the sequence of matrices obtained by the RCA. Our approachis based on Theorem 2.16 in [7], which states that: Let M be a compact simply connected manifold of dimension m > . Given a finitely generated free chain complex C with H ∗ ( C ) ≈ H ∗ ( M ) , then there exists a self-indexingMorse function g on M such that if M k = g − (( −∞ , k + 1 / then the chain complex { H k ( M k , M k − ) , ∂ k } isisomorphic to C . Let { (cid:101) ∆ r } mr =0 be the sequence of matrices produced when one applies the Row Cancellation to ∆. For each r ,denote by (cid:98) ∆ r the matrix obtained from (cid:101) ∆ r by removing the rows and columns ( p + 1) and ( p − ξ + 1), for eachprimary pivot (cid:101) ∆ rp − ξ +1 ,p +1 in the ξ -th diagonal of (cid:101) ∆ r , for ξ = 1 , . . . , r . Observe that the last matrix (cid:98) ∆ m is null,since all non zero entries of (cid:101) ∆ m are above a primary pivot.Now, for each r = 1 , . . . , m , consider the pair ( C ( r ) , (cid:98) ∆ r ) where C ( r ) is generated by the subset of Crit ( f )consisting of all the critical points of f except for the ones cancelled in the previous step ( r − C ( r ) , (cid:98) ∆ r ) is a chain complex whose homology coincides with the homology of M . In fact, it follows fromTheorems 18 and 28 that the primary pivots marked during RCA are ± 1, hence, each change of basis in RCAis a change of basis over Z . Moreover, in order to construct C ( r ), only pairs of cancelling critical points areremoved and since these do not correspond to generators of H ( M ), then the homology of ( C ( r ) , (cid:98) ∆ r ) coincideswith the singular homology of M .By Theorem 2.6 in [7], there is a self-indexing Morse function g r on M such that the chain complex { H k ( M k ( r ) , M k − ( r )) , ∂ rk } is isomorphic to ( C ( r ) , (cid:98) ∆ r ) , where M k ( r ) = g − r (( −∞ , k + 1 / { H k ( M k ( r ) , M k − ( r )) , ∂ rk } is in fact a Morse chain complex for the function g r , see [1]. Consider the Morse flow ϕ r associated to the vector field −∇ g r . Moreover, since the last matrix (cid:101) ∆ m produced by the RCA generates The map ∂ k : H k ( M k , M k − ) → H k − ( M k − , M k − ) is the boundary map of the triple ( M k , M k − , M k − ). 38 null matrix (cid:98) ∆ m , then the flow ϕ m corresponds to the gradient flow associated to a perfect Morse function g m . (cid:3) In this section, we will consider the case where M is a surface. A surface connection matrix has some additionalproperties which enable us to prove stronger dynamical results. In this case, each algebraic cancellation of themodules of the spectral sequence, codified by the Spectral Sequence Sweeping Algorithm, can be interpreted asa dynamical cancellation.Surfaces connection matrices are endowed with special features unique to the two-dimensional case satisfyingthe following properties:(i) ∆ ij ∈ { , , − } ;(ii) columns and rows of ∆ may be partitioned into three groups, namely J , J and J , the first associatedwith wells ( h ’s), the second with saddles ( h ’s) and the third with sources ( h ’s). Figure 7 shows thetypical structure of grouped surface connection matrix. Block ∆ J J contains the connections from saddlesto wells, while block ∆ J J contains the connections from sources to saddles;(iii) each column in ∆ J J contains either two nonzero elements, namely 1 and − 1, or none;(iv) each row in ∆ J J contains either two non zero elements, namely 1 and − 1, or none;or is obtained from a matrix with the properties above by multiplying a subset of rows and/or columns by − J (cid:122) (cid:125)(cid:124) (cid:123) J (cid:122) (cid:125)(cid:124) (cid:123) J (cid:122) (cid:125)(cid:124) (cid:123) ∆ = ∆ J J ∆ J J (cid:122)(cid:125)(cid:124)(cid:123) J (cid:122)(cid:125)(cid:124)(cid:123) J (cid:122)(cid:125)(cid:124)(cid:123) J Figure 7: Grouped surface connection matrix with J = { , . . . , } , J = { , . . . , } and J = { , . . . , } .The proof of this characterization for surfaces connection matrices can be found in [3]. This characterizationallows us to prove the following result. Lemma 29 A surface connection matrix is TU. roof . It suffices to show that a matrix satisfying (i)–(iv) is TU, since this property is invariant under multipli-cation of rows and/or columns by − J J (resp., ∆ J J ) is TU because it is a 0, ± M = ∆ LC be a square submatrix of ∆. We showby induction on the order of M that its determinant is 0, ± 1. The statement is trivially true when M is 1 × n × n submatrices and consider a submatrix of order n +1. Let L k = L ∩ J k and C k = C ∩ J k ,for k = 0, 1, 2. If L (cid:54) = ∅ or C (cid:54) = ∅ , the determinant of M will be zero, and the statement will be true. Sosuppose L = C = ∅ . If L = ∅ (resp., L = ∅ , C = ∅ , C = ∅ ), then M is a submatrix of ∆ J . (resp., ∆ J . ,∆ . J , ∆ . J ), a TU matrix, and its determinant is 0, ± 1. So the only case that remains to be analyzed is when L , L , C and C are all nonempty. Notice that M L C and M L C are null.Consider the submatrix M L C . If all its columns contain two non zeros, then the sum of the rows in M L . is zero and det M = 0, proving the assertion. If not, there is a column that is either null, implying M has a nullcolumn and thus null determinant, or contains exactly one non zero entry, say m ij . Since M L C is zero, we canapply Laplace’s expansion on this column and express det M as ± m ij times the determinant of the submatrix (cid:102) M of M obtained by removing row i and column j therefrom. But the induction hypothesis may be applied to (cid:102) M , since it is of order n , and thus prove the assertion for the remaining case. (cid:3) Lemma 29 implies Theorem 18 is valid for surface connection matrices. Corollary 30 (Primary pivots for orientable surfaces) Let ∆ be a surface connection matrix. Then theprimary pivots obtained when applying the Incremental Sweeping Algorithm over F thereto value ± . As consequence of Corollary 30, every non zero differential is an isomorphism and hence it produces aalgebraic cancellation of the modules of the spectral sequence. Our approach in [3] was to interpret the algebraiccancellation of the modules of the spectral sequence, which has been coded by the Spectral Sequence SweepingAlgorithm, as dynamical cancellations.Whenever a dynamical cancellation occurs, all the connecting orbits to both singularities cancelled mustdisappear immediately. The Row Cancellation Algorithm reflects exactly this situation and hence it is bettersuited to relate dynamical interpretation to algebraic cancellations. When the input to the Row CancellationAlgorithm over F is restricted to the special class of surface connection matrices, it is called Smale’s CancellationSweeping Algorithm .It follows from Lemma 29 and Theorem 28 that, for a surface connection matrix, the primary pivots markedin both the Sweeping Algorithm over Z and Smale’s Cancellation Sweeping Algorithm coincide in value andposition. Corollary 31 (Equality of primary pivots for orientable surfaces) Let ∆ ∈ {− , , } m be a surface con-nection matrix with column/row partition J , J , J . The primary pivots marked in the application of the Sweep-ing Algorithm over Z to ∆ coincide in value and position with the primary pivots marked in the application ofSmale’s Cancellation Sweeping Algorithm thereto. We reproduce Theorem 5 . heorem 32 Let ( C, ∆) be the Morse chain complex associated to a Morse-Smale function f . Let ( E r , d r ) bethe associated spectral sequence for the finest filtration F = { F p C } defined by f . The algebraic cancellation ofthe modules E r of the spectral sequence are in one-to-one correspondence with dynamical cancellations of criticalpoints of f . The proof of this theorem, sketched out in [3], relies on Corollary 30 and Corollary 31 proved in this article.In conclusion, Theorem 28 is crucial in producing the association of the algebraic cancellation of modules of aspectral sequence of a filtered chain complex and the dynamical cancellation of the critical points that generatesthis chain complex. See [3] for more details. Final Remarks In this article as well as in [4, 5], we have explored the algebraic tool provided by the spectral sequence and itsdynamical implications. This was a major step in investigating the dynamics associated to the spectral sequence.In particular, we obtined results on the efect the change of basis of the connection matrices had on the changesin generators of the modules E rp which were coded in the connection matrices determined by the SSSA in termsof a continuation of the initial Morse decomposition. Many questions arise in this algebraic-dynamical setting.The work developed in [4] for spectral sequences ( E r , d r ) where each E rp is a Z -module admits the possibilityof modules with torsion which may disappear through some algebraic cancellation as one calculates the spec-tral sequence or remain after the sequence stabilizes. What is the dynamical meaning of the torsion in thesetwo different cases? Note that torsion never appears for TU matrices. The following example illustrates thisphenmena.Inspired by the results in [3] and in this article, where the algebra has its dynamical correspondence and ismostly determined by the primary pivots, we mean to explore in the case aforementioned, primary and change-of-basis pivots which are not necessarily equal to ± 1. Our motivation for this investigation is that certain primaryand change-of-basis pivots correspond to nonzero differentials of the spectral sequence as proved in [4]. (cid:45) (cid:45) (cid:45) (cid:45) 33 2 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) 33 2 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) 33 2 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) 33 5 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) 33 5 2 4 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) 22 0 0 0 0 0 0 0 0 0 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) As we can see in the example, as r increases, the Z -module E rp changes generators and the SSSA brings thisabout when a change-of-basis pivot is marked, causing a change of basis over Q of the matrix ∆ r in order todetermine the connection matrix ∆ r +1 of the next stage. Note that in the example, even though the entries inthe intermediary matrices can be fractional, the primary pivots and change-of basis pivots are always integers.Furthermore, the entries of the last matrix are all integers. This is true in general and was proved in [5].However, many questions remain. Up to what point does the SSSA associate a continuation of the initial flowto the unfolding of the algebra? What is the dynamical meaning of the matrices of the intermediary stages? Forinstance, what do the rational entries mean? 43 [ h (2) k − ] Z [ h (3) k − ] Z [ h (4) k ] Z [ h (5) k ] Z [ h (6) k ] Z [ h (7) k ] Z [ h (8) k ] Z [ h (9) k ] Z [ h (10) k +1 ] Z [ h (11) k +1 ] Z [ h (12) k +1 ] Z [ h (13) k +1 ] Z [ h (2) k − ] Z [ h (3) k − ] 0 Z [ h (5) k ] Z [ h (6) k ] Z [ h (7) k ] Z [ h (8) k ] Z [ h (9) k ] 0 Z [ h (11) k +1 ] Z [ h (12) k +1 ] Z [ h (13) k +1 ] Z [ h (2) k − ] 0 0 Z [2 h (5) k − h (4) k ] Z [ h (6) k ] Z [ h (7) k ] Z [ h (8) k ] Z [ h (9) k ] 0 Z [ h (11) k +1 + h (10) k +1 ] Z [ h (12) k +1 ] Z [ h (13) k +1 ] Z [ h (2) k − ] 0 0 Z [2 h (5) k − h (4) k ] Z [ h (6) k − h (5) k + h (4) k ] Z [ h (7) k ] Z [ h (8) k ] 0 0 Z [ h (11) k +1 + h (10) k +1 ] Z [2 h (12) k +1 − h (10) k +1 ] Z [ h (13) k +1 ]0 0 0 Z [2 h (5) k − h (4) k ] 0 Z [ h (7) k ] 0 0 0 0 0 Z [ h (13) k +1 + h (12) k +1 − h (10) k +1 ]0 0 0 Z [2 h (5) k − h (4) k ] 0 Z [ h (7) k − h (6) k + h (5) k − h (4) k ] 0 0 0 0 0 Z [ h (13) k +1 + h (12) k +1 − h (10) k +1 ]0 0 0 Z [2 h (5) k − h (4) k ] 0 0 0 0 0 0 0 Z [5 h (13) k +1 +5 h (12) k +1 + h (11) k +1 − h (10) k +1 ]0 0 0 Z [2 h (5) k − h (4) k ] 0 0 0 0 0 0 0 Z [5 h (13) k +1 +5 h (12) k +1 + h (11) k +1 − h (10) k +1 ]0 0 0 Z [2 h (5) k − h (4) k ] 0 0 0 0 0 0 0 0 − − − E E E E E E E E E References [1] A. 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