Computing the Hermite Form of a Matrix of Ore Polynomials
aa r X i v : . [ c s . S C ] O c t Computing the Hermite Form of a Matrix ofOre Polynomials
Mark Giesbrecht
Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada
Myung Sub Kim
Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada
Abstract
Let F [ ∂ ; σ, δ ] be the ring of Ore polynomials over a field (or a skew field) F , where σ is anautomorphism of F and δ is a σ -derivation. Given a matrix A ∈ F [ ∂ ; σ, δ ] m × n , we show howto compute the Hermite form H of A and a unimodular matrix U such that UA = H . Thealgorithm requires a polynomial number of operations in F in terms of the dimensions m and n ,and the degrees (in ∂ ) of the entries in A . When F = k ( z ) for some field k , it also requires timepolynomial in the degrees in z of the coefficients of the entries, and if k = Q it requires timepolynomial in the bit length of the rational coefficients as well. Explicit analyses are provided forthe complexity, in particular for the important cases of differential and shift polynomials over Q ( z ). To accomplish our algorithm, we apply the Dieudonn´e determinant and quasideterminanttheory for Ore polynomial rings to get explicit bounds on the degrees and sizes of entries in H and U .
1. Introduction
The Ore polynomials are a natural algebraic structure which captures difference, q -difference, differential, and other non-commutative polynomial rings. The basic conceptsof pseudo-linear algebra are presented nicely by Bronstein and Petkovˇsek (1996); see(Ore, 1931) for the seminal introduction.On the other hand, canonical forms of matrices over commutative principal idealdomains (such as Z or F [ x ], for a field F ) have proven invaluable for both mathematicaland computational purposes. One of the successes of computer algebra over the pastthree decades has been the development of fast algorithms for computing these canonicalforms. These include triangular forms such as the Hermite form (Hermite, 1851), low Email addresses: [email protected] (Mark Giesbrecht), [email protected] (Myung Sub Kim).
Preprint submitted to Elsevier 1 November 2012 egree forms like the Popov form (Popov, 1972), as well as the diagonal Smith form(Smith, 1861).Canonical forms of matrices over non-commutative domains, especially rings of dif-ferential and difference operators, are also extremely useful. These have been examinedat least since the work of Dickson (1923), Wedderburn (1932), and Jacobson (1943).Recently they have found uses in control theory (Chyzak, Quadrat, and Robertz, 2005;Zerz, 2006; Hal´as, 2008). Computations with multidimensional linear systems over Orealgebras are nicely developed by Chyzak, Quadrat, and Robertz (2007), and an excellentimplementation of many fundamental algorithms is provided in the OreModules packageof Maple.In this paper we consider canonical forms of matrices of Ore polynomials over a skewfield F . Let σ : F → F be an automorphism of F and δ : F → F be a σ -derivation.That is, for any a, b ∈ F , δ ( a + b ) = δ ( a ) + δ ( b ) and δ ( ab ) = σ ( a ) δ ( b ) + δ ( a ) b . We thendefine F [ ∂ ; σ, δ ] as the set of usual polynomials in F [ ∂ ] under the usual addition, but withmultiplication defined by ∂a = σ ( a ) ∂ + δ ( a )for any a ∈ F . This is well-known to be a left (and right) principal ideal domain, with astraightforward Euclidean algorithm (see (Ore, 1933)).Some important cases over the field of rational functions F = k ( z ) over a field k are asfollows:(1) σ ( z ) = S ( z ) = z + 1 is a so-called shift automorphism of k ( z ), and δ identically zeroon k . Then k ( z )[ ∂ ; S ,
0] is generally referred to as the ring of shift polynomials . Witha slight abuse of notation we write k ( z )[ ∂ ; S ] for this ring.(2) δ ( z ) = 1 and σ ( z ) = z , so δ ( h ( z )) = h ′ ( z ) for any h ∈ k ( z ) with h ′ its usual derivative.Then k ( z )[ ∂ ; σ, δ ] is called the ring of differential polynomials . With a slight abuse ofnotation we write k ( z )[ ∂ ; ′ ] for this ring.A primary motivation in the definition of k ( z )[ ∂ ; ′ ] is that there is a natural ac-tion on the space of infinitely differentiable functions in z , namely the differentialpolynomial a m ∂ m + a m − ∂ m − + · · · + a ∂ + a ∈ k ( z )[ ∂ ; ′ ]acts as the linear differential operator a m ( z ) d m f ( z ) dz m + a m − ( z ) d m − f ( z ) dz m − + · · · + a ( z ) df ( z ) dz + a ( z ) f ( z )on an infinitely differentiable function f ( z ). See (Bronstein and Petkovˇsek, 1996).The (row) Hermite form we will compute here is achieved purely by row operations,and we treat a matrix A ∈ F [ ∂ ; σ, δ ] m × n as generating the left F [ ∂ ; σ, δ ]-module of itsrows. Thus, by left row rank , we mean the rank of the free left F [ ∂ ; σ, δ ]-module of rowsof A , and will denote this simply as the rank of A for the remainder of the paper. Amatrix H ∈ F [ ∂ ; σ, δ ] m × n of rank r is in Hermite form if an only if (i)
Only the first r rows are non-zero; (ii) In each row the leading (first non-zero) element is monic; (iii)
All entries in the column below the leading element in any row are zero; (iv)
All entries in the column above the leading element in any row are of lower degreethan the leading element. 2or square matrices of full rank the Hermite form will thus be upper triangular withmonic entries on the diagonal, whose degrees dominate all other entries in their column.For example, in the differential polynomial ring Q ( z )[ ∂ ; ′ ] as above: A = + ( z +2) ∂ + ∂ + (2 z +1) ∂ + (1+ z ) ∂ (2 z + z ) + z ∂ (2+2 z +2 z ) + ∂ z + z (3+ z ) + (3+ z ) ∂ + ∂ z ) + (5+3 z ) ∂ + ∂ z ) + (2+4 z ) ∂ ∈ Q ( z )[ ∂ ; ′ ] × (1.1)has Hermite form H = (2+ z ) + ∂ z − z +2 z z − z ∂ (2+ z ) + ∂ + z + ∂ − z + − z + z z ∂ + ∂ ∈ Q ( z )[ ∂ ; ′ ] × . Note that the Hermite form may have denominators in z . Also, while this example doesnot demonstrate it, the degrees in the Hermite form, in both numerators and denomina-tors in z and ∂ , are generally substantially larger than in the input (in Theorem 5.6 wewill provide polynomial, though quite large, bounds on these degrees, and suspect thesebounds may well be met generically).For any matrix A ∈ F [ ∂ ; σ, δ ] n × n of full rank, there exists a unique unimodular matrix U ∈ F [ ∂ ; σ, δ ] n × n (i.e., a matrix whose inverse exists and is also in F [ ∂ ; σ, δ ] n × n ) suchthat U A = H is in Hermite form. This form is canonical in the sense that if two matrices A, B ∈ F [ ∂ ; σ, δ ] n × n are such that A = P B for unimodular P ∈ F [ ∂ ; σ, δ ] n × n then theHermite form of A equals the Hermite form of B . Existence and uniqueness of the Hermiteform are established much as they are over Z n × n in Section 2. For rank deficient matricesand rectangular matrices, the Hermite form also exists but the transformation matrixmay not be unique. See Section 6 for further details.In commutative domains such as Z and F [ x ] there have been enormous advances in thepast two decades in computing Hermite, Smith and Popov forms. Polynomial-time algo-rithms for the Smith and Hermite forms over F [ x ] were developed by Kannan (1985), withimportant advances by Kaltofen, Krishnamoorthy, and Saunders (1987), Villard (1995),Mulders and Storjohann (2003), Pernet and Stein (2010), and many others. One of thekey features of this recent work in computing canonical forms has been a careful analysisof the complexity in terms of matrix size, entry degree, and coefficient swell. Clearlyidentifying and analyzing the cost in terms of all these parameters has led to a dramaticdrop in both theoretical and practical complexity.Computing the classical Smith and Hermite forms of matrices over Ore domainshas received less attention though canonical forms of differential polynomial matriceshave applications in solving differential systems and control theory (see (Hal´as, 2008;Kotta, Leiback, and Hal´as, 2008)). Abramov and Bronstein (2001) analyze the numberof reduction steps necessary to compute a row-reduced form , whileBeckermann, Cheng, and Labahn (2006) analyze the complexity of row reduction interms of matrix size, degree and the sizes of the coefficients of some shifts of the inputmatrix. Beckermann et al. (2006) demonstrate tight bounds on the degree and coeffi-cient sizes of the output, which we will employ here. For the Popov form, Cheng (2003)gives an algorithm for matrices of shift polynomials. Cheng’s approach involves order3ases computation in order to eliminate lower order terms of Ore polynomial matrices.A main contribution of Cheng (2003) is to give an algorithm computing the rank and arow-reduced basis of the left nullspace of a matrix of Ore polynomials in a fraction-freeway. This idea is extended in Davies, Cheng, and Labahn (2008) to compute the Popovform of general Ore polynomial matrices. They reduce the problem of computing Popovform to a nullspace computation. However, though Popov form is useful for rewritinghigh order terms with respect to low order terms, we want a different canonical formmore suited to solving system of linear diophantine equations. Since the Hermite form isupper triangular, it meets this goal nicely, not to mention the fact that it is a “classical”canonical form. An implementation of the basic (exponential-time) Hermite algorithm isprovided by Culianez (2005). In (Giesbrecht and Kim, 2009), we present a polynomial-time algorithm for the Hermite form over Q ( z )[ ∂ ; ′ ], for full rank square matrices. Whileit relies on similar techniques as this current paper, the cost of the algorithm is higher, thecoefficient bounds weaker, and it does not work for matrices of general Ore polynomials.The related “two-sided” problem of computing the Jacobson (non-commutative Smith)canonical form has also been recently considered. Blinkov, Cid, Gerdt, Plesken, and Robertz(2003) implement the standard algorithm in the package Janet . Levandovskyy and Schindelar(2011) provide a very complete implementation, for the full Ore case over skew fields, ofa Jacobson form algorithm using Gr¨obner bases in Singular. Middeke (2008) has recentlydemonstrated that the Jacobson form of a matrix of differential polynomials can be com-puted in time polynomial in the matrix size and degree (but the coefficient size is notanalyzed). Giesbrecht and Heinle (2012) give a probabilistic polynomial-time algorithmfor this problem in the differential case.One of the primary difficulties in both developing efficient algorithms for matrices ofOre polynomials, and in their analysis, is the lack of a standard notion of determinant, andthe important bounds this provides on degrees in eliminations. In Section 3 we establishbounds on the degrees of entries in the inverse of a matrix over any non-commutative fieldwith a reasonable degree function. We do this by introducing the quasideterminant ofGel’fand and Retakh (1991, 1992) and analyzing its interaction with the degree function.We also prove similar bounds on the degree of the Dieudonn´e determinant. In both cases,the bounds are essentially the same as for matrices over a commutative function field.In Section 4 we consider matrices over the Ore polynomials and bound the degrees ofentries in the Hermite form and corresponding unimodular transformation matrices. Wealso bound the degrees of the Dieudonn´e determinants of these matrices.In Section 5 we present our algorithm for the Hermite form. The degree bounds andcosts of our algorithms are summarized as follows, from Theorems 4.7, 4.9 and 5.6.
Summary Theorem.
Let A ∈ k [ z ][ ∂ ; σ, δ ] n × n have full rank with entries of degree atmost d in ∂ , and coefficients of degree at most e in z . Let H ∈ k ( z )[ ∂ ; σ, δ ] n × n be theHermite form of A and U ∈ k ( z )[ ∂ ; σ, δ ] n × n such that U A = H .(a) The sum of degrees in ∂ in any row of H is at most nd , and each entry in U hasdegree in ∂ at most ( n − d .(b) All coefficients from k ( z ) of entries of H and U have degrees in z , of both numeratorsand denominators, bounded by O ( n de ).(c) We can compute H and U deterministically with O ˜( n d e ) † , operations in k . † We employ soft-Oh notation: for functions σ and ϕ we say σ ∈ O ˜( ϕ ) if σ ∈ O ( ϕ log c ϕ ) for someconstant c ≥ k has at least 4 n de elements. We can compute the Hermite form H and U with an expected number of O ˜( n d e ) of operations in k using standard polynomialarithmetic. This algorithm is probabilistic of the Las Vegas type; it never returnsan incorrect answer.The cost of our algorithm for Ore polynomials over an arbitrary skew field, as wellas over more specific fields like Q ( z ), is also shown to be polynomially bounded, and isdiscussed in Section 5.It should be noted from the above theorem that the output is of quite substantial size.The transformation matrix U as above is an n × n matrix of polynomials in ∂ of degreebounded by nd in ∂ and each coefficient has degree bounded by n de , for a total size of O ( n d e ) elements of k . While we have not proven our size bounds are tight, we havesome confidence they are quite strong.The algorithm presented in Section 5 is derived from the “linear systems” approachof Kaltofen et al. (1987) and Storjohann (1994). In particular, it reduces the problem tothat of linear system solving over the generally commutative ground field (e.g., k ( z )).There are efficient algorithms and implementations for solving this problem. While weexpect that further algorithmic refinements and reductions in cost can be achieved be-fore an industrial-strength implementation is made, the general approach of reducingto well-studied computational problem in a commutative domain would seem to haveconsiderable merit in theory and practice.In Section 6 we show that for the case of rank-deficient and rectangular matrices, thecomputation of the Hermite form is reduced to the full rank, square case.
2. Existence and Uniqueness of the Hermite form over Ore domains
In this section we establish the basic existence and uniqueness of Hermite forms overOre domains. These follow similarly to the traditional proofs over Z ; see for example(Newman, 1972, Theorems II.2 and II.3), which we outline below. Fact 2.1 (Jacobson (1943), Section 3.7) . Let a, b ∈ F [ ∂ ; σ, δ ] , not both zero with g =gcrd( a, b ) , u, v ∈ F [ ∂ ; σ, δ ] such that ua + vb = g , and s, t ∈ F [ ∂ ; σ, δ ] such that sa = − tb = lclm( a, b ) . Then W = u vs t ∈ F [ ∂ ; σ, δ ] × is such that W ab = g , and W is unimodular. This is easily generalized to n × n matrices as follows. Lemma 2.2.
Let w = ( w , . . . , w n ) T ∈ F [ ∂ ; σ, δ ] n × , and i, j ∈ { , . . . , n } . There existsa matrix E = E ( i, j ; w ) ∈ F [ ∂ ; σ, δ ] n × n such that Ew = ( u , . . . , u n ) T ∈ F [ ∂ ; σ, δ ] n × , with u i = gcrd( w i , w j ) and u j = 0 . roof. If both w i = w j = 0 are zero, then E is the identity matrix. If w i = 0 and w j = 0, then let E be the permutation matrix which swaps rows j and i .Otherwise, let W = u vs t be as in Fact 2.1 with ( a, b ) T = ( w i , w j ) T , and W ( w i , w j ) T =( g, T for g = gcrd( w i , w j ). Define E ( i, j ; w ) as the identity matrix except E ii = u, E ij = v, E ji = s, E jj = t. Clearly E satisfies the desired properties. ✷ We note that off diagonal entries in a triangular matrix can be unimodularly reducedby the diagonal entry below it.
Lemma 2.3.
Let J ∈ F [ ∂ ; σ, δ ] n × n be upper triangular with non-zero diagonal. Thereexists a unimodular matrix R ∈ F [ ∂ ; σ, δ ] n × n , which is upper triangular and has oneson the diagonal, such that in every column of RJ , the degree of each diagonal entry isstrictly larger than the degrees of the entries above it. Proof.
For any a, b ∈ F [ ∂ ; σ, δ ] with b = 0, we have a = qb + r for quotient q ∈ F [ ∂ ; σ, δ ]and remainder r ∈ F [ ∂ ; σ, δ ] with deg ∂ r < deg ∂ b , and − q ab = rb . Embedding such unimodular matrices Q into n × n identity matrices, we can “reduce”the off diagonal entries of J by the diagonal entries below them. ✷ Theorem 2.4.
Let A ∈ F [ ∂ ; σ, δ ] n × n have full rank. Then there exists a matrix H ∈ F [ ∂ ; σ, δ ] n × n in Hermite form, and a unimodular matrix U ∈ F [ ∂ ; σ, δ ] n × n , such that U A = H . Proof.
The proof follows by observing the traditional (but inefficient) algorithm to com-pute the Hermite form. We first use a (unimodular row) permutation to move any non-zero element in column 1 into the top left position; failure to find a non-zero element incolumn 1 means our matrix is rank deficient. We then repeatedly apply Lemma 2.2 tofind Q such that Q A only has the top left position non-zero. This same procedure isthen repeated on subdiagonal of columns 2 , , . . . , n in sequence, so there exists a uni-modular matrix Q = Q · · · Q n such that QA is upper triangular. The matrix is thenunimodularly reduced using Lemma 2.3. ✷ Theorem 2.5.
Let A ∈ F [ ∂ ; σ, δ ] n × n have full row rank. Suppose U A = H for unimodular U ∈ F [ ∂ ; σ, δ ] n × n and Hermite form H ∈ F [ ∂ ; σ, δ ] n × n . Then both U and H are unique. roof. Suppose H and G are both Hermite forms of A . Thus, there exist unimodularmatrices U and V such that U A = H and V A = G , and G = W H where W = V U − is unimodular. Since G and H are upper triangular matrices, we know W is as well.Moreover, since G and H have monic diagonal entries, the diagonal entries of W equal1. We now prove W is the identity matrix. By way of contradiction, first assume that W is not the identity, so there exists an entry W ij which is the first nonzero off-diagonalentry on the i th row of W . Since i < j and since W ii = 1, G ij = H ij + W ij H jj . Because W ij = 0, we see deg ∂ G ij ≥ deg ∂ G jj , which contradicts the definition of the Hermiteform.Uniqueness of U is easily established since U A = V A , so U − V = I and U = V . ✷
3. Non-commutative determinants and degree bounds for linear equations
One of the main difficulties in matrix computations in skew (non-commutative) fields,and a primary difference with the commutative case, is the lack of the usual determinant.In particular, the determinant allows us to bound the degrees of solutions to systems ofequations, the size of the inverse or other decompositions, not to mention the degreesat intermediate steps of computations, through Hadamard-like formulas and Cramer’srules.The most common non-commutative determinant was defined by Dieudonn´e (1943),and is commonly called the
Dieudonn´e determinant . It preserves some of the multiplica-tive properties of the usual commutative determinant, but is insufficient to establishthe degree bounds we require (amongst other inadequacies). Gel’fand and Retakh (1991,1992) introduced quasideterminants and a rich associated theory as a central tool in lin-ear algebra over non-commutative rings. Quasideterminants are more akin to the (inverseof the) entries of the classical adjoint of a matrix than a true determinant. We employquasideterminants here to establish bounds on the degree of the entries in the inverse ofa matrix, and on the Dieudonn´e determinant in this section, and on the Hermite formand its multiplier matrices in Section 4.We will establish bounds on degrees of quasideterminants and Dieudonn´e determinantsfor a general skew field K with a degree deg : K → Z ∪ {−∞} satisfying the followingproperties. For a, b ∈ K :(i) If a = 0 then deg a ∈ Z , and deg 0 = −∞ ;(ii) deg( a + b ) ≤ max { deg a, deg b } ;(iii) deg( ab ) = deg a + deg b ;(iv) If a = 0 then deg( a − ) = − deg a .As a simple commutative example, if K = F ( y ) for some field F and commutingindeterminate y , for any a = a N /a D with polynomials a N , a D ∈ F [ y ] ( a D = 0), we candefine deg a = deg a N − deg a D .More properly, our degree function is a non-archimedean valuation on K . Since ourmain application will be to non-commutative Ore polynomial rings, where degrees are anatural and traditional notion, we will adhere to the nomenclature of degrees. We note,however, that the degrees as defined here may become negative.
See Lemma 4.3 for theeffective application to the Ore polynomial case.7 .1. Quasideterminants and degree bounds
Following Gel’fand and Retakh (1991, 1992), we define the quasideterminant as a col-lection of n functions from K n × n → K ∪ {⊥} , where ⊥ represents the function being undefined . Let A ∈ K n × n and p, q ∈ { , . . . , n } . Assume A pq ∈ K is the ( p, q ) entry of A ,and let A ( pq ) ∈ K ( n − × ( n − be the matrix A with the p th row and q th column removed.Define the ( p, q )-quasideterminant of A as | A | pq = A pq − X i = p,j = q A pi ( | A ( pq ) | ji ) − A jq , where the sum is taken over all summands where | A ( pq ) | ji is defined. If all summands have | A ( pq ) | ji undefined then | A | pq is undefined (and has value ⊥ ). See (Gel’fand and Retakh,1992). Fact 3.1 (Gel’fand and Retakh (1991), Theorem 1.6) . Let A ∈ K n × n over a (possiblyskew) field K .(1) The inverse matrix B = A − ∈ K n × n exists if and only if the following are true:(a) If the quasideterminant | A | ij is defined then | A | ij = 0 , for all i, j ∈ { , . . . , n } ;(b) For all p ∈ { , . . . , n } there exists a q ∈ { , . . . , n } , such that the quasidetermi-nant | A | pq is defined;(c) For all q ∈ { , . . . , n } there exists a p ∈ { , . . . , n } such that the quasidetermi-nant | A | pq is defined;(2) If the inverse B exists, then for i, j ∈ { , . . . , n } we have B ji = ( ( | A | ij ) − if | A | ij is defined, if | A | ij is not defined. Over a commutative field K , where A ∈ K n × n has inverse B , the quasideterminantsbehave like a classical adjoint: | A | ij = ( − i + j det A/ det A ( ij ) = 1 /B ji . If B ji is zerothen | A | ij is undefined.We now bound the size of the quasideterminants in terms of the size of the entries of A . Assume that K has a degree function as above. Theorem 3.2.
Let A ∈ K n × n , such that either A ij = 0 or ≤ deg A ij ≤ d for all i, j ∈ { , . . . , n } . For all p, q ∈ { , . . . , n } such that | A | pq is defined we have − ( n − d ≤ deg | A | pq ≤ nd . Proof.
We proceed by induction on n .For n = 1, p = q = 1 and | A | = A , so clearly the property holds. Assume thestatement is true for dimension n −
1. Thendeg | A | pq = deg A pq − X i = p,j = q A pi ( | A ( pq ) | ji ) − A jq , where the sum is over all defined summands. Then using the inductive hypothesis wehave deg | A | pq ≤ max (cid:26) deg A pq , max i = p,j = q n deg A pi − deg | A ( pq ) | ji + deg A jq o(cid:27) ≤ d + ( n − d ≤ nd, | A | pq ≥ − deg | A ( pq ) | ji ≥ − ( n − d. ✷ Corollary 3.3.
Let A ∈ K n × n be unimodular, and B ∈ K n × n such that AB = I . Assume A ij = 0 or ≤ deg A ij ≤ d for all i, j ∈ { , . . . , n } . Then deg B ≤ ( n − d . Proof.
From Fact 3.1 we know that B ji = ( | A | ij ) − when | A | ij is defined (and B ji = 0otherwise). Thus deg B ji = − deg | A | ij ≤ ( n − d , and B ij = 0 or deg B ij ≥ A isunimodular. ✷ Let [ K ∗ , K ∗ ] be the commutator subgroup of the multiplicative group K ∗ of K , the(normal) subgroup of K ∗ generated by all pairs of elements of the form a − b − ab for a, b ∈ K ∗ . Thus K ∗ / [ K ∗ , K ∗ ] is a commutative group.Let A ∈ K n × n be a matrix with a right inverse. The Bruhat Normal Form of A isa decomposition A = T DP V , where P ∈ K n × n is a permutation matrix inducing thepermutation σ : { , . . . , n } → { , . . . , n } , and T, D, V ∈ K n × n are T = ∗ · · · ∗ · · · ∗ ... . . . ...0 · · · , D = diag( u , . . . , u n ) , V = · · · ∗ · · · ∗ · · · ∗ . See (Draxl, 1983, Chapter 19) for more details. The Bruhat decomposition arises fromGaussian elimination, much as the
LU P decomposition does in the commutative case.We then define δετ ( A ) = sign( σ ) · u · · · u n ∈ K (sometimes called the pre-determinant of A ). Let π be the canonical projection from K ∗ → K / [ K ∗ , K ∗ ]. Then the Dieudonn´edeterminant is defined as D et( A ) = π ( δετ ( A )) ∈ K / [ K ∗ , K ∗ ], or D et( A ) = 0 if A is notinvertible.The Dieudonn´e determinant has a number of the desirable properties of the usualdeterminant, as proven in (Dieudonn´e, 1943):(1) D et( AB ) = D et( A ) D et( B ) for any A, B ∈ K n × n ;(2) D et( P ) = 1 for any permutation matrix;(3) D et A C B = D et( A ) D et( B ).Also note that if K has a degree function as above, then deg( D et( A )) is well defined,since all elements of the equivalence class of π ( D et( A )) have the same degree (since thedegree of all members of the commutator subgroup is zero). Gel’fand and Retakh (1991)show that δετ ( A ) = | A | | A (11) | | A (12 , | | A (123 , | · · · | A (1 ...n − , ...,n − | nn = | A | · δετ ( A (11) ) , P is the identity in theBruhat decomposition above), where A (1 ...k, ...k ) is the matrix A with rows 1 . . . k andcolumns 1 . . . k removed (keeping the original labelings of the remaining rows and columns).More generally, let R = ( r , . . . , r n ), C = ( c , . . . , c n ) be permutations of { , . . . , n } ,let R k = ( r , . . . , r k ), C k = ( c , . . . , c k ), and define A ( R k ,C k ) as the matrix A with rows r , . . . , r k and columns c , . . . , c k removed (where A ( R ,C ) = A ). Define δετ R,C ( A ) = | A | r ,c | A ( R ,C ) | r ,c | A ( R ,C ) | r ,c · · · | A ( R n − ,C n − ) | r n ,c n = | A | r ,c · δετ R,C ( A ( r ,c ) )= | A | r ,c · | A ( R ,C ) | r ,c · δετ ( A ( R ,C ) ) . (3.1) Fact 3.4 (Gelfand, Gelfand, Retakh, and Wilson (2005), Section 3.1) . Let
R, C be per-mutations of { , . . . , n } and R k , C k defined as above. If | A ( R k ,C k ) | r k +1 ,c k +1 is defined for k = 0 . . . n − , then D et( A ) = sign( R ) · sign( C ) · π ( δετ R,C ( A )) . In other words, the Dieudonn´e determinant is essentially invariant of the order of thesequence of submatrices specified in (3.1).
Theorem 3.5.
Let A ∈ K n × n be invertible, with deg A ij ≤ d . Then deg D et( A ) ≤ nd . Proof.
We proceed by induction on n . For n = 1 this is clear. For n = 2, the possiblepredeterminants are δετ , ( A ) = | A | A = ( A − A A − A ) A ,δετ , ( A ) = | A | A = ( A − A A − A ) A ,δετ , ( A ) = | A | A = ( A − A A − A ) A ,δετ , ( A ) = | A | A = ( A − A A − A ) A , at least one of which must be defined and non-zero, and all of which clearly have degreeat most 2 d .Now assume the theorem is true for matrices of dimension less than n . Choose r , c ∈{ , . . . , n } such that | A | r ,c is non-zero and of minimal degree; that is deg | A | r ,c ≤ deg | A | k,ℓ for all k, ℓ such that | A | k,ℓ is defined and non-zero. The fact that | A | r ,c = 0implies that A ( r ,c ) is invertible, and we can continue this process recursively. Thus,let R = ( r , . . . , r n ) and C = ( c , . . . , c n ) be permutations of { , . . . , n } such that | A ( R i ,C i ) | r i +1 ,c i +1 = 0 and deg | A ( R i ,C i ) | r i +1 ,c i +1 is minimal over the degrees of non-zero,10efined quasideterminants | A ( R i ,C i ) | k,ℓ , for 0 ≤ i < n . Now δετ R,C ( A ) = | A | r ,c · | A ( r ,c ) | r ,c · δετ R,C ( A ( R ,C ) )= A r ,c − X k,ℓ A r k | A ( r ,c ) | − ℓk A ℓc · | A ( r ,c ) | r ,c · δετ R,C ( A ( R ,C ) )= A r ,c · | A ( r ,c ) | r ,c · δετ R,C ( A ( R ,C ) ) − X k,ℓ A r k | A ( r ,c ) | − ℓk A ℓc · | A ( r ,c ) | r ,c · δετ R,C ( A ( R ,C ) )= A r ,c · δετ R,C ( A ( R ,C ) ) − X k,ℓ A r k | A ( r ,c ) | − ℓk A ℓc · | A ( r ,c ) | r ,c · δετ R,C ( A ( R ,C ) ) , where all sums are taken only over defined quasideterminants as above. Thusdeg D et( A ) = deg δετ R,C ( A ) ≤ max { d + ( n − d, d + ( n − d } ≤ nd, using the induction hypothesis and the assumption that deg | A ( r ,c ) | r ,c is chosen to beminimal. ✷
4. Degree bounds on matrices over F [ ∂ ; σ, δ ] Some well-known properties of F [ ∂ ; σ, δ ] are worth recalling; see (Ore, 1933) for theoriginal theory or (Bronstein and Petkovˇsek, 1994) for an algorithmic presentation. Given f, g ∈ F [ ∂ ; σ, δ ], there is a degree function (in ∂ ) which satisfies the usual properties:deg ∂ ( f g ) = deg ∂ f + deg ∂ g and deg ∂ ( f + g ) ≤ max { deg ∂ f, deg ∂ g } . We set deg ∂ −∞ . F [ ∂ ; σ, δ ] is a left (and right) principal ideal ring, which implies the existence of aright (and left) division with remainder algorithm such that there exists unique q, r ∈ F [ ∂ ; σ, δ ] such that f = qg + r where deg ∂ ( r ) < deg ∂ ( g ). This allows for a right (and left)Euclidean-like algorithm which shows the existence of a greatest common right divisor, h = gcrd( f, g ), a polynomial of minimal degree (in ∂ ) such that f = uh and g = vh for u, v ∈ F [ ∂ ; σ, δ ]. The GCRD is unique up to a left multiple in F \{ } , and thereexist co-factors a, b ∈ F [ ∂ ; σ, δ ] such that af + bg = gcrd( f, g ). There also exists a leastcommon left multiple lclm( f, g ). Analogously there exists a greatest common left divisor,gcld( f, g ), and least common right multiple, lcrm( f, g ), both of which are unique up toa right multiple in F . From (Ore, 1933) we also have thatdeg ∂ lclm( f, g ) = deg ∂ f + deg ∂ g − deg ∂ gcrd( f, g ) , deg ∂ lcrm( f, g ) = deg ∂ f + deg ∂ g − deg ∂ gcld( f, g ) . (4.1)It will be useful to work in the quotient skew field F ( ∂ ; σ, δ ) of F [ ∂ ; σ, δ ], and to extendthe degree function deg ∂ appropriately. We first show that any element of F ( ∂ ; σ, δ ) canbe written as a standard fraction f g − , for f, g ∈ F [ ∂ ; σ, δ ] (and in particular, since F [ ∂ ; σ, δ ] is non-commutative, we insist that g − is on the right). Fact 4.1 (Ore (1933), Section 3) . Every element of F ( ∂ ; σ, δ ) can be written as a standardfraction. F ( ∂ ; σ, δ ) as follows. Definition 4.2.
For f, g ∈ F [ ∂ ; σ, δ ], g = 0, the degree deg ∂ ( f g − ) = deg ∂ f − deg ∂ g .The proof of the next lemma is left to the reader. Lemma 4.3.
For f, g, u, v ∈ F [ ∂ ; σ, δ ] , with g, v = 0 , we have the following:(a) if f g − = uv − then deg ∂ ( f g − ) = deg ∂ ( uv − ) ;(b) deg ∂ (( f g − ) · ( uv − )) = deg ∂ ( f g − ) + deg ∂ ( uv − ) ;(c) deg ∂ ( f g − + uv − ) ≤ max { deg ∂ ( f g − ) , deg ∂ ( uv − ) } ;(d) deg ∂ (( f g − ) − ) = − deg ∂ ( f g − ) . In summary, the degree function on F ( ∂ ; σ, δ ) meets the requirement of a degree func-tion on a skew field as in Section 3, and is once again, actually a valuation on F ( ∂ ; σ, δ ). We show unimodular matrices are precisely those with a Dieudonn´e determinant ofdegree zero.
Lemma 4.4.
Let W ∈ F [ ∂ ; σ, δ ] × be as in Fact 2.1. Then deg ∂ D et W = 0 . Proof.
We may assume that gcrd( a, b ) = g = 1, since the same matrix satisfies W ( ag − , bg − ) T = (1 , T . Also assume both a, b = 0 (otherwise the lemma is trivial).Then u vs t a b = v t , and D et( W ) · a ≡ t mod [ F [ ∂ ; σ, δ ] ∗ , F [ ∂ ; σ, δ ] ∗ ] , so deg ∂ D et W + deg ∂ a = deg ∂ t . Since gcrd( a, b ) = 1, from (4.1) we know deg ∂ a = deg ∂ t ,so deg ∂ D et W = 0. ✷ Embedding the 2 × n × n identity matrices, as in Lemma 2.2, we obtainthe following (the proof of which is left to the reader). Corollary 4.5.
Let E ∈ F [ ∂ ; σ, δ ] n × n be as in Lemma 2.2. The deg ∂ D et E = 0 . The characterization of unimodular matrices as those with Dieudonn´e determinant ofdegree zero follows by looking at the Hermite form of a unimodular matrix.
Theorem 4.6. U ∈ F [ ∂ ; σ, δ ] n × n is unimodular if and only if deg ∂ D et U = 0 . Proof.
Suppose U is unimodular. The Hermite form of U must be the identity: all thediagonal entries must be invertible in F [ ∂ ; σ, δ ] and the entries above the diagonal arereduced to 0. Thus, the unimodular multiplier to the Hermite form of U will be theinverse U .Following the simple algorithm to compute the Hermite form in Theorem 2.4, we see itworked via a sequence of unimodular transforms, all of which were either permutations,off-diagonal reductions from Lemma 2.2, or are of the form E in Lemmas 2.2 and 4.5. The12ieudonn´e determinants of permutations and reduction transformations are both equalto 1, by the basic properties of Dieudonn´e determinants discussed at the beginning ofSection 3.2, and hence have degree 0. The Dieudonn´e determinants of the transformations E are of degree 0 by Corollary 4.5. The proof is now complete by the multiplicativeproperty of Dieudonn´e determinants, and the additive properties of their degrees.Assume conversely that deg ∂ D et U = 0, and that V ∈ F [ ∂ ; σ, δ ] n × n is a unimodu-lar matrix such that V U = H is in Hermite form. Then deg ∂ D et V + deg ∂ D et U =deg ∂ D et H = 0. But the only matrix in Hermite form with degree 0 is the identitymatrix. Thus V is the inverse of U , and U must be unimodular. ✷ In this section we establish degree bounds on Hermite forms of matrices over F [ ∂ ; σ, δ ]and their unimodular transformation matrices. Theorem 4.7.
Let A ∈ F [ ∂ ; σ, δ ] n × n have full rank and entries of degree at most d andHermite form H ∈ F [ ∂ ; σ, δ ] n × n . Then(a) The sum of the degrees of the diagonal entries of H has degree at most nd ;(b) The sum of the degrees of the entries in any row of H has degree at most nd . Proof.
Let V ∈ F [ ∂ ; σ, δ ] n × n be unimodular such that A = V H , whence D et( A ) = D et( V ) D et( H ). Therefore (a) follows fromdeg ∂ D et( A ) = deg ∂ D et( H ) = X ≤ i ≤ n deg ∂ H ii ≤ nd. Point (b) follows from the fact that each entry above the diagonal in the Hermite formhas, by definition, degree smaller than the degree of the diagonal entry below it. ✷ We now show that all entries in H − have non-positive degrees. Lemma 4.8.
Let H ∈ F [ ∂ ; σ, δ ] n × n be of full rank and in Hermite form, and let J = H − .Then deg ∂ J ij ≤ for ≤ i, j ≤ n . Proof.
We consider the equation JH = I , and note that J , like H is upper triangular.For each r ∈ { , . . . , n } we show by induction on c (for r ≤ c ≤ n ) that deg ∂ J rc ≤ c = r , J rr H rr = 1, so deg ∂ J rr = − deg ∂ H rr ≤ r < c and deg ∂ J rℓ ≤ r ≤ ℓ < c . We need to show thatdeg ∂ J rc ≤
0. We know that X ≤ i ≤ n J rℓ H ℓc = X r ≤ ℓ ≤ c J rℓ H ℓc = 0 . Since deg ∂ J rℓ ≤ r ≤ ℓ < c and deg ∂ H cc > deg ∂ H ℓc for r ≤ ℓ < c , it must be thecase that deg ∂ J rc ≤ ✷ Theorem 4.9.
Let A ∈ F [ ∂ ; σ, δ ] n × n be invertible (over F ( ∂ ; σ, δ ) ), whose entries all havedegree at most d in ∂ . Suppose A has Hermite form H ∈ F [ ∂ ; σ, δ ] n × n , with U A = H and U V = I for U, V ∈ F [ ∂ ; σ, δ ] n × n . Then deg ∂ V ≤ d and deg ∂ U ≤ ( n − d . roof. Note that V = AH − , and by Lemma 4.8 all entries in H − have non-positivedegree. Thus deg ∂ V ≤ deg ∂ A . By Corollary 3.3, deg ∂ U ≤ ( n − d . ✷
5. Computing Hermite forms by linear systems over F [ ∂ ; σ, δ ] In this section we present our polynomial-time algorithm to compute the Hermiteform of a matrix over F [ ∂ ; σ, δ ]. This generally follows the “linear systems” approach ofKaltofen et al. (1987), and more specifically the refinements in Storjohann (1994) (formatrices over k [ x ] for a field k ). We will need the tools for F [ ∂ ; σ, δ ] we have developedin the previous sections. The method only directly constructs the matrix U such that H = U A . The Hermite form H can be found by performing the multiplication U A .The general approach is similar to that described in Giesbrecht and Kim (2009), with aprimary difference that in that paper the technique of Kaltofen et al. (1987) was adapted.This new technique is considerably more efficient (see below). As well, our earlier paperwas constrained to differential rings as the necessary degree bounds were not availablefor all Ore polynomials.Assume that A ij = P ≤ k ≤ d A ijk ∂ k for A ijk ∈ F . Let row( A, i ) ∈ F [ ∂ ; σ, δ ] × n be the i th row of A and define L ( A ) = X ≤ i ≤ n b i · row( A, i ) : b , . . . , b n ∈ F [ ∂ ; σ, δ ] , the left module of the row space of A . The following lemma is shown analogously to(Storjohann, 1994, § Lemma 5.1.
Let A ∈ F [ ∂ ; σ, δ ] n × n be nonsingular, with Hermite form H . Let h i =deg ∂ H ii for ≤ i ≤ n . For v = (0 , . . . , , v ℓ , . . . , v n ) ∈ F [ ∂ ; σ, δ ] × n , with deg ∂ v ℓ < h ℓ ,then if v ∈ L ( A ) we have v ℓ = 0 , and if v ℓ = 0 then v / ∈ L ( A ) . The following theorem is analogous to (Storjohann, 1994, § Theorem 5.2.
Let A ∈ F [ ∂ ; σ, δ ] n × n have full rank, with deg ∂ A ij ≤ d for ≤ i, j ≤ n .Let ( d , . . . , d n ) be a given vector of non-negative integers. Let T be an n × n matrix with T ij = P ≤ k ≤ ̺ t ijk ∂ k for unknowns t ijk , where ̺ ≥ ( n − d + max i { d i − h i } . Considerthe system of equations in t ijk with constraints: ( T A ) i,i,d i = 1 , for ≤ i ≤ n , — diagonal entries are monic;( T A ) i,i,k = 0 , for k > d i , — diagonal entry in row i has degree d i ;( T A ) i,j,k = 0 , for i = j and k ≥ d j — off diagonal entries have lower degreethan the diagonal entry in that column. (5.1) By a solution for T we mean an assignment of variables t ijk ← α ijk ∈ F for some ≤ i, j ≤ n and ≤ k ≤ ̺ such (5.1) holds.Let h , . . . , h n ∈ N be the degrees of the diagonal entries of the Hermite form of A .The following statements about the above system hold:(i) If d i ≥ h i for ≤ i ≤ n then there exists a solution for T ; ii) If there exists a positive integer ℓ ≤ n such that d i = h i for ≤ i < ℓ and d ℓ < h ℓ then there is no solution for T ;(iii) If d i = h i for ≤ i ≤ n then there is a unique solution for T such that G = T A isequal to the Hermite form of A under that solution. Proof.
Let H ∈ F [ ∂ ; σ, δ ] n × n be the Hermite form of A and let U ∈ F [ ∂ ; σ, δ ] n × n be theunique unimodular matrix such that U A = H .To show (i), let D = diag( ∂ d − h , . . . , ∂ d n − h n ) ∈ F [ ∂ ; σ, δ ] n × n , and consider theequality DU A = DH . Let H ∗ ∈ F [ ∂ ; σ, δ ] n × n be the Hermite form of DH and U ∗ ∈ F [ ∂ ; σ, δ ] n × n the unimodular matrix such that U ∗ DU A = H ∗ . We construct H ∗ from DH simply by reducing the entries above the diagonal (since it is already upper triangular).Thus U ∗ is upper triangular, with ones on the diagonal, and deg ∂ U ∗ ij < ( d i − h i ) − ( d j − h j )for i < j . We claim T = U ∗ DU is a solution to (5.1) . First note that the particularchoice of D , together with the definition of H ∗ ensure that the constraints of (5.1) aremet. Furthermore, entries in the i th row of U ∗ D have degree at most d i − h i . By Theorem4.9, deg ∂ U ≤ ( n − d , hence deg ∂ T ≤ ( n − d + max i { d i − h i } ≤ ̺ .To prove (ii), suppose by contradiction that there exists a nonnegative integer ℓ ≤ n and a solution T such that deg ∂ (( T A ) ii ) = d i for 1 ≤ i < ℓ and deg ∂ (( T A ) ℓℓ ) < h ℓ . Notethat row( T A, ℓ ) = ((
T A ) ℓ, , . . . , ( T A ) ℓ,n ) is in L ( A ). First, if ℓ = 1, then deg ∂ ( T A ) ℓ,
1. Then deg ∂ ( T A ) ℓ, < h (to satisfy (5.1)), and hence byLemma 5.1, so ( T A ) ℓ, = 0. A simple induction shows that ( T A ) ℓ,j = 0 for 1 ≤ j < ℓ .Now consider ( T A ) ℓ,ℓ , which has degree d ℓ < h ℓ by our assumption. Again by Lemma5.1 ( T A ) ℓ,ℓ = 0, which (5.1) ensures is monic, a contradiction.If the conditions of (iii) hold, then by (i) there exists at least one solution for T . Wecan use an inductive proof similar to that used in our proof of (ii) to show that elementsbelow the diagonal in T A are zero (i.e., that (
T A ) ij = 0 for i > j ). By the uniqueness ofthe Hermite form we must have T A = H . ✷ This theorem allows us to work with a partial order on the degree sequences. Forany ( h , . . . , h n ) , ( d , . . . , d n ) ∈ Z n , we say that ( h , . . . , h n ) (cid:22) ( d , . . . , d n ) if and onlyif h i ≤ d i for all 1 ≤ i ≤ n (and similarly define (cid:22) for strict precedence). Thus, (5.1)has a solution if and only if ( h , . . . , h n ) (cid:22) ( d , . . . , d n ) and this is unique if and only if( h , . . . , h n ) = ( d , . . . , h n ).We now embed the system (5.1) into a system of linear equations over F , with no Orecomponent. We embed F [ ∂ ; σ, δ ] into vectors over F via τ ℓ : F [ ∂ ; σ, δ ] → F ℓ +1 , with τ ℓ ( u + u ∂ + u ∂ + · · · + u ℓ ∂ ℓ − ) = ( u , . . . , u ℓ ) ∈ F ℓ +1 . For g ∈ F [ ∂ ; σ, δ ] of degree d , u ∈ F [ ∂ ; σ, δ ] of degree at most m , and assuming ℓ ≥ m + d ,the equation ug = f can be realized by a matrix equation over F :( u , . . . , u m ) τ ℓ ( g ) τ ℓ ( ∂g )... τ ℓ ( ∂ m g ) | {z } µ ℓm ( g ) ∈ F ( m +1) × ( ℓ +1) = ( f , . . . , f ℓ ) ⇐⇒ τ m ( u ) µ ℓm ( g ) = τ ℓ ( f ) . d , . . . , d n ∈ N as in Theorem 5.2, and setting ̺ ≥ ( n − d + max i { d i − h i } ,we can then study (5.1), as realized as (a subset of) the linear equations in the matrixequation over F : τ ̺ ( T ) · · · τ ̺ ( T n )... ... τ ̺ ( T n ) · · · τ ̺ ( T nn ) | {z }b T ∈ F n × ( ̺ +1) n µ ̺ + d̺ ( A ) · · · µ ̺ + d̺ ( A n )... ... µ ̺ + d̺ ( A n ) · · · µ ̺ + d̺ ( A nn ) | {z }b A ∈ F n ( ̺ +1) × ( ̺ + d +1) n = τ ̺ + d ( G ) · · · τ ̺ + d ( G n )... ... τ ̺ + d ( G n ) · · · τ ̺ + d ( G nn ) . | {z }b G ∈ F n × ( ̺ + d +1) n (5.2)This set of equations is a superset of the equation (5.1). Some entries in b G are unknown,in particular those corresponding to coefficients of degrees (in ∂ ) strictly less than thedegree of the diagonal below it. However, these entries in b G are not mentioned or involvedin Theorem 5.2, and we can remove these columns from b G . Similarly, since they imposeno constraint on (5.1), we can remove the corresponding columns of b A . By Theorem 5.2,if we know d , . . . , d n , the remaining equations will have a unique solution, from whichwe completely determine b T . Example 5.3.
Consider the following matrix in Q ( z )[ ∂ ; ′ ] (the differential polynomialsover Q ( z )): A = ( z + 1) + ∂ z + z∂ ∂ ( z + z ) + z∂ z + 1 2 ∂ ( − z − z ) − z∂ z∂ z∂ ∈ Q ( z )[ ∂ ; ′ ] × . Assume for this example that we know the degrees of the entries in the Hermite form are( d , d , d ) = (1 , , n = 3, and we can set ̺ = 2, and have b T z }| { t t t t t t t t t t t t t t t t t t t t t t t t t t t b A z }| { z + 1 1 0 0 z z z + 1 1 0 1 z + 1 z z + 1 1 0 2 z + 2 z z + z z z + 1 0 0 0 0 2 0 02 z + 1 z + z + 1 z z + 1 0 0 0 0 2 02 4 z + 2 z + z + 2 z z + 1 0 0 0 0 2 − z − z − z z z − z − − z − z − − z z z − − z − − z − z − − z z z = b G z }| { G G G G G G G G G .
16s noted above, b G still has some indeterminates, from columns which specify coef-ficients of the entries of G which are of degree strictly less than the maximum in thecorresponding column of G . These entries are not mentioned in (5.1), and we removethem to form e G . The corresponding columns of A are similarly not involved in (5.1),and are removed to form e A . We obtain now obtain a reduced system of equations whichcorresponds precisely to (5.1): b T z }| { t t t t t t t t t t t t t t t t t t t t t t t t t t t e A z }| { z z z + 1 1 0 1 z + 1 z z + 1 1 0 2 z + 2 z z z + 1 0 0 0 0 0 z + z + 1 z z + 1 0 0 2 04 z + 2 z + z + 2 z z + 1 0 0 2 − z z − z − z − − z z z − z − − z − z − − z z z = e G z }| { ∈ F n ( ̺ +1) × n ( ̺ +1) . By Theorem 5.2, since we have “guessed” the degree sequence of the diagonal entries( d , d , d ) = (1 , ,
2) correctly, the system has a unique solution: b T = z +12 z +1 − z z +1 − z +12 z +1 − z z +1 z +12 z +1 z z +1 − z +3 z +2( z + z +2)(2 z +1) − zz + z +2 z + z − z + z +2)(2 z +1) z +1 z + z +2 z − z − z − z ( z + z +2)(2 z +1) zz + z +2 which corresponds to T = z +12 z +1 − z z +1 − z +12 z +1 − z z +1 z +12 z +1 z z +1 − z +3 z +2( z + z +2)(2 z +1) − zz + z +2 ∂ z + z − z + z +2)(2 z +1) + z +1 z + z +2 ∂ z − z − z − z ( z + z +2)(2 z +1) ) + zz + z +2 ∂ ∈ Q ( z )[ ∂ ; ′ ] × giving H = T A = ( z + 1) + ∂ − z +2 z − z +1 ∂ z + z +22 z +1 ∂ z +3 z − z − z + z +2)(2 z +1) ∂ + ∂ ∈ Q ( z )[ ∂ ; ′ ] ×
17n Hermite form.We can now state our algorithm for computing the Hermite form given the degrees ofthe diagonal elements.
Algorithm
HermiteFormGivenDegrees
Input: A ∈ F [ ∂ ; σ, δ ] n × n of full rank, with (unknown) Hermite form H with diagonaldegrees ( h , . . . , h n ) ∈ N n ; Input: ( d , . . . , d n ) ∈ N n , the proposed degrees of the diagonal entries of H Output: H ∈ F [ ∂ ; σ, δ ] n × n if ( d , . . . , d n ) = ( h , . . . , h n ), or a message that ( d , . . . , d n )is lexicographically smaller or larger than ( h , . . . , h n ); Let ̺ = ( n − d + max i d i Form the matrix equation b T b A = b G as in (5.2) Remove all columns from b G containing an indeterminate, and corresponding columnsfrom b A , to form the “reduced” linear system b T e A = e G , where e A and e G are nowmatrices over F if rank e A < ( n + 1) ̺ then return “( h , . . . , h n ) (cid:22) ( d , . . . , d n )” // System is underconstrained if b T e A = e G has no solution then return “( h , . . . , h n ) (cid:14) ( d , . . . , d n )” // System is inconsistent Solve the system b T e A = e G for b T Construct T ∈ F [ ∂ ; σ, δ ] n × n from b T return H = T A and U = T From Theorem 4.7 we know that each entry in the Hermite form of A ∈ F [ ∂ ; σ, δ ] n × n ,with deg ∂ A ij ≤ d for 1 ≤ i, j ≤ n , has degree at most nd . If the diagonal entries of A have degrees ( h , . . . , h n ), then we know that(0 , . . . , (cid:22) ( h , . . . , h n ) (cid:22) ( nd, nd, . . . , nd ) . Algorithm
HermiteFormGivenDegrees detects whether our choice of degree sequence isequal to, larger than, or not larger than or equal to the actual one. Thus, a simplecomponent-wise binary search allows us to find the actual degree sequence ( h , . . . , h n ).That is, start by finding for the h by executing HermiteFormGivenDegrees with degreesequence ( d , nd, . . . , nd ) for different values of d . This will require O (log( nd )) attempts.Then search for h using degree sequence O ( h , d , nd, . . . , nd ) for different values of d , etc. It will require at most O ( n log( nd )) attempts to find the entire correct degreesequence ( h , . . . , h n ). Lemma 5.4.
Given A ∈ F [ ∂ ; σ, δ ] n × n of full rank, where each entry has degree (in ∂ ) lessthan d , we can compute the Hermite form H ∈ F [ ∂ ; σ, δ ] n × n of A , and U ∈ F [ ∂ ; σ, δ ] n × n such that U A = H . The algorithm requires us to call Algorithm HermiteFormGivenDegrees O ( n log( nd )) times, with input A and varying degree sequences. For a first, general analysis of the complexity we will assume that operations in F haveunit cost (and hence no coefficient growth is accounted for). To perform the rank testin Step 4, the inconsistency test in Step 6, and the equation solution in Step 8, we cansimply do an LU decomposition of e A using Gaussian elimination. e A has size n ( ̺ + 1) × m ,18here n ( ̺ + 1) ≤ m ≤ n ( ̺ + d + 1), i.e., O ( n d ) × O ( n d ). Gaussian elimination, whichcomputes an LU -decomposition or more generally a Bruhat normal form (see Section3.2 or (Draxl, 1983, Chapter 19)) is effective over any skew field, and on a p × q matrixrequires O ( p q ) operations, and hence in our case can be accomplished with O ( n d )operations in F . Combining this with Lemma 5.4 we obtain the following. Theorem 5.5.
Let A ∈ F [ ∂ ; σ, δ ] n × n have full rank with entries of degree (in ∂ ) lessthan d . We can compute the Hermite form H ∈ F [ ∂ ; σ, δ ] n × n of A , and U ∈ F [ ∂ ; σ, δ ] n × n such that U A = H . The algorithm requires O ( n d log( nd )) operations in F . We next analyze our algorithm for computing the Hermite form of a matrix A ∈ k ( z )[ ∂ ; σ, δ ] n × n over the field F = k ( z ), where k is a field and z an indeterminate. Withoutloss of generality A ∈ k [ z ][ ∂ ; σ, δ ] n × n by clearing denominators (which is a left-unimodularoperation), but note that the Hermite form may still be in k ( z )[ ∂ ; σ, δ ] (see Example5.3). We will also assume for convenience that σ ( z ) ∈ k [ z ] and deg z δ ( z ) ≤
1. Thus ∂z = σ ( z ) ∂ + δ ( z ) ∈ k [ z ][ ∂ ] and the degree in z and ∂ remains unchanged. A moregeneral analysis could follow similarly.We assume that multiplying two polynomials in k [ z ] of degree at most m can be ac-complished with O ( M ( m )) operations in k : M ( m ) = m using standard arithmetic or M ( m ) = m log m log log m using fast arithmetic (Cantor and Kaltofen, 1991). We simi-larly assume that two integers with ℓ bits can be multiplied with O ( M ( l )) bit operations.Finally, when we talk of the degree of a rational function in k ( z ) we mean the maxi-mum degree of the numerator and denominator, assuming they are co-prime. This givesa reasonable indication of representation size. Theorem 5.6.
Let A ∈ k [ z ][ ∂ ; σ, δ ] n × n have full rank with entries of degree at most d in ∂ , and of degree at most e in z . Let H ∈ k ( z )[ ∂ ; σ, δ ] n × n be the Hermite form of A and U ∈ k ( z )[ ∂ ; σ, δ ] n × n such that U A = H .(a) deg z H ij ∈ O ( n de ) and deg z U ij ∈ O ( n de ) for ≤ i, j ≤ n .(b) We can compute H and U deterministically with O ( n d log( nd ) · M ( n de )) or O ˜( n d e ) operations in k .(c) Assume k has at least n de elements. We can compute the Hermite form H and U with an expected number of O ( n d log( nd ) + n d e ) of operations in k usingstandard polynomial arithmetic. This algorithm is probabilistic of the Las Vegastype; it never returns an incorrect answer. Proof.
To show (a), recall that the matrix e A is of size O ( n d ) × O ( n d ) and degree O ( e ). Using Hadamard’s bound and Cramer’s rule, the numerators and denominators in b T thus have degree at most O ( n de ) in z . H = U A has the same degree bound in z .To prove (b) we solve the system of equations (5.1) as in Theorem 5.5 but now takinginto account coefficient growth. Since we have an explicit bound on the degree in z ofnumerators and denominators of the solution, we can compute modulo an irreduciblepolynomial Γ ∈ k [ z ] more than twice this degree and recover the solution over k ( z ) byrational recovery. Each operation in k ( z ) ∈ k [ z ] / (Γ) will thus take O ˜( M ( n de )) operationsin k . The stated total cost follows from the cost in Theorem 5.5 multiplied by thisoperation cost.To show (c), we note that the tests for rank deficiency in Step 4, and inconsistencyin Step 6, can be done by considering the equation b T e A = e G mod ( z − α ) for a randomly19hosen α from a subset of k of size at least 4 n de . This follows because the largest invariantfactor w ∈ k [ z ] of e A has degree at most n de by Hadamard’s bound (see part (a)), andthe rank modulo ( z − α ) changes only if α is a root of w . By the Schwartz-Zippel Lemma(Schwartz, 1980) this happens with probability at most 1 / α (andthis probability of error can be made exponentially smaller by repeating with differentrandom choices). Thus, these tests require only O ( n d ) operations in k to perform,correctly with high probability. During the binary search for the degree sequence we onlyperform these cheaper tests, requiring a total of O ( n d log( nd )) operations in k beforefinding the correct degree sequence.Once we have found the correct degree sequence, we employ Dixon’s (1982) algorithmto solve the linear system over k ( z ) (this is the fastest known algorithm using standardmatrix arithmetic, and is very effective in practice; one could also employ the asymptoti-cally faster method of Storjohann (2003) with sub-cubic matrix arithmetic). This lifts thesolution to the system modulo ( z − α ) i for i = 1 , . . . , n de , where α is a non-root of the(unknown) largest invariant factor of A (i.e., is such that rank A = rank A mod ( z − α )).Computing the solution modulo ( z − α ) n de is sufficient to recover the solution in k ( z )using rational function reconstruction, since both the numerator and denominator havedegree less than n de by part (a); see (von zur Gathen and Gerhard, 2003), Section 5.7.A random choice of α from a subset of k of size 4 n de is sufficient to obtain a non-zero ofthe largest invariant factor (and hence not change the dimension of the solution space)with probability at least 1 / A mod ( z − α ) using O ( n d ) operationsin k . We then lift the solution to b T e A ≡ e G mod ( z − α ) i for i = 0 , . . . , n de . Each liftingstep requires O ( n d ) operations in k , yielding a total cost of O ( n d e ). ✷ For comparison, the cost of the algorithm in (Giesbrecht and Kim, 2009), for the caseof matrices over k ( z )[ ∂ ; ′ ], required O ˜( n d e ) operations in k .Finally, we consider coefficient growth in Q of Ore polynomial rings over Q ( z ). For thecomputation, once we have constructed the matrix e A , we can bound the coefficient-sizesin b T directly using Hadamard-type bounds. We can then employ a Chinese remainderscheme to find the Hermite form using the above algorithm (or any other method, for thatmatter). For example, we could simply choose a single prime p with twice as many bits asthe largest numerator or denominator in the solution to (5.2) and then compute modulothat prime, in Z p [ z ]; the rational coefficients of H can be recovered by integer rationalreconstruction from their images in Z p (von zur Gathen and Gerhard, 2003, § A ∈ Z [ z ][ ∂ ; σ, δ ] n × n without loss of generality.For convenience in this analysis (though not in complete generality), we assume thatdeg z δ ( z ) ≤ σ ( z ) ∈ Z [ z ], so ∂z = σ ( z ) ∂ + δ ( z ) ∈ Z [ z ].For a polynomial a = a + a z + · · · + a m z m ∈ Z [ z ], let k a k ∞ = max i | a i | . For f = f ( z ) + f ( z ) ∂ + · · · + f r ( z ) ∂ r ∈ Z [ z ][ ∂ ; σ, δ ], let k f k ∞ = max i k f i k ∞ . Define k A k ∞ =max ij k A ij k ∞ . In equation (5.2), the entries in e A have size at most k A k ( ̺ ) ∞ = max ij max ℓ {k A ij k ∞ , k ∂A ij k ∞ , . . . , k ∂ ̺ A ij k ∞ } ∈ Z . (5.3)20 heorem 5.7. Let A ∈ Z [ z ][ ∂ ; σ, δ ] n × n be of full rank and such that deg ∂ ( A ) = d , deg z ( A ) ≤ e and k A k ( ̺ ) ∞ ≤ β . Then the Hermite form H ∈ Q ( z )[ ∂ ; σ, δ ] n × n and U ∈ Q ( z )[ ∂ ; σ, δ ] n × n such that U A = H satisfy log k H k ∞ , log k U k ∞ ∈ O ( n d (log e + log β + log n + log d )) . Proof.
Entries in e A are polynomials in Z [ z ] of degree at most e and coefficient sizeat most β . Every minor of e A , and hence each entry in the solution b T , is bounded byHadamard’s bound, which in this case is (cid:0) (1 + e ) β ( n d ) (cid:1) O ( n d ) (see Giesbrecht (1993) Theorem 1.5 for height bounds on polynomial products). ✷ By performing all computations modulo an appropriately large prime (as discussedabove), we immediately get the following.
Corollary 5.8.
Let A ∈ Z [ z ][ ∂ ; σ, δ ] n × n have full rank with entries of degree at most d in ∂ , of degree at most e in z , and k A k ( ̺ ) ∞ ≤ β (where ̺ = O ( n d ) ). Let H ∈ Q ( z )[ ∂ ; σ, δ ] n × n be the Hermite form of A and U ∈ Q ( z )[ ∂ ; σ, δ ] n × n such that U A = H .We can compute the Hermite form H ∈ Q ( z )[ ∂ ; σ, δ ] n × n of A , and U ∈ Q ( z )[ ∂ ; σ, δ ] n × n such that U A = H , using an algorithm that requires an expected number O (( n d log( nd )+ n d e ) · M ( n d (log e + log β + log n + log d ))) , or O ˜( n d e log β ) bit operations. This al-gorithm is probabilistic of the Las Vegas type (never returning an incorrect answer). The following corollary summarizes this growth explicitly over two common rings, thedifferential polynomials Q ( z )[ ∂ ; ′ ], and the shift polynomial Q ( z )[ ∂ ; S ]. Corollary 5.9.
Let A ∈ Z [ z ][ ∂ ; σ, δ ] n × n be of full rank and such that deg ∂ ( A ) = d , deg z ( A ) ≤ e , H ∈ Q ( z )[ ∂ ; σ, δ ] n × n the Hermite form of A , and U ∈ Q ( z )[ ∂ ; σ, δ ] n × n suchthat U A = H . For both the differential polynomials Q ( z )[ ∂ ; ′ ] (where σ ( z ) = z , δ ( z ) = 1 )and the shift polynomials Q ( t )[ ∂ ; S ] (where σ ( z ) = z + 1 , δ ( z ) = 0 ), we have log k U k ∞ , log k H k ∞ ∈ O ˜( n d ( e + log k A k ∞ )) . Proof.
To show this for differential polynomials, we note that for a = P ≤ i ≤ d a i ( z ) ∂ i ∈ Z [ z ][ ∂ ; ′ ], ∂ ℓ a = X ≤ j ≤ ℓ (cid:18) ℓj (cid:19) X ≤ i ≤ d a i ( z ) ( j ) ∂ ℓ − j , where a i ( z ) ( j ) is the j th derivative of a i ( z ). Since only the first e derivatives of any a i are non-zero k ∂ ℓ a k ∞ ≤ ℓ e · k a k ∞ · e !and hence log k A k ( ̺ ) ∞ ∈ O (log k A k ∞ + e log( n d )) for ̺ = O ( n d ). The result follows byTheorem 5.7.To show this for shift polynomials we note that for a = P ≤ i ≤ d a i ( z ) ∂ i ∈ Z [ ∂, S ], ∂ ℓ a = X ≤ i ≤ d a i ( z + ℓ ) ∂ i , k ∂ ℓ a k ∞ ≤ k a k ∞ e/ ℓ e , and hence log k A k ( ̺ ) ∞ ∈ O (log k A k ∞ + e log( n d )) for ̺ = n d . Again, the result followsfrom Theorem 5.7. ✷
6. Rectangular and rank deficient matrices
To this point we have assume that our matrices A ∈ F [ ∂ ; σ, δ ] n × n were both squareand of full rank. In this section we relax both these conditions to show a polynomial-timealgorithm for the Hermite form in all cases. Suppose now that A ∈ F [ ∂ ; σ, δ ] m × n has rank r ≤ m . We first show how to compute aunimodular matrix P ∈ F [ ∂ ; σ, δ ] n × n such that P A has precisely r non-zero rows. Since P is unimodular, the left F [ ∂ ; σ, δ ]-modules generated by the rows of A and the rows of P A are equal.We employ the row reduction algorithm developed by Beckermann et al. (2006), anddiscussed in (Davies et al., 2008). Let b = m · deg ∂ A and Q = diag( ∂ b , . . . , ∂ b ) ∈ F [ ∂ ; σ, δ ] n × n , and form the matrix B = AQ − I n ∈ F [ ∂ ; σ, δ ] ( m + n ) × n . We compute a basis for the left nullspace basis of B in Popov form using the algorithmin Beckermann et al. (2006), and suppose it has form (cid:16) P | R (cid:17) ∈ F [ ∂ ; σ, δ ] m × ( m + n ) for P ∈ F [ ∂ ; σ, δ ] m × m . Then by (Davies et al., 2008, Theorem 4.5, 5.5), P ∈ F [ ∂ ; σ, δ ] m × m is unimodular and P A is in Popov form. In particular, only r rows are non-zero. Theorem 6.1.
Let A ∈ k [ z ][ ∂ ; σ, δ ] n × n have rank r ≤ m , with deg ∂ A ≤ d and deg z A ≤ e . We can find a unimodular matrix P ∈ k [ z ][ ∂ ; σ, δ ] m × m such that P A has r non-zerorows using O ( m n (deg ∂ A ) (deg z A ) ) operations in k . It also requires a polynomialnumber of bit operations when k = Q . Proof.
The matrix B has deg ∂ ( B ) ≤ ( m + 1) · deg ∂ A and deg z ( B ) ≤ deg z ( A ). Thealgorithm of Beckermann et al. (2006) to compute the Popov form, as summarized intheir Corollary 7.7, requires O ( m n (deg ∂ A ) (deg z A ) ) operations in k . It also requiresa polynomial number of bit operations when k = Q . ✷ After eliminating the zero rows, we are left to compute the Hermite form of a (possiblyrectangular) matrix with full rank. 22 .2. Rectangular matrices of full rank
Let A ∈ F [ ∂ ; σ, δ ] m × n have full rank with n > m . Then there exists a lexicographicallyfirst set of columns τ , . . . , τ m such that the submatrix A τ = A ,τ A ,τ · · · A ,τ m ... ... A m,τ A m,τ · · · A m,τ m ∈ F [ ∂ ; σ, δ ] m × m has full rank. We can compute the unique U such that U A τ is in Hermite form using thealgorithm of the previous section. Then it must be the case that U A has form
U A = H ,τ ∗ · · · · · · · · · ∗ H ,τ ∗ · · · ∗ . . . H m,τ m ∈ F [ ∂ ; σ, δ ] m × n , the Hermite form of A . This suggests an easy strategy of simply conducting a binarysearch for the lexicographically smallest subset { τ , . . . , τ m } of { , . . . , n } such that the U computed to put A τ in Hermite form also puts U A in Hermite form. Since there are (cid:0) nm (cid:1) ≤ n subsets of size m , our binary search will require at most log (cid:0) nm (cid:1) ≤ n iterations.We can now offer the following theorem for rectangular matrices of full row rank. Theorem 6.2.
Let A ∈ k [ z ][ ∂ ; σ, δ ] m × n have full rank with entries of degree at most d in ∂ , and of degree at most e in z . Let H ∈ k ( z )[ ∂ ; σ, δ ] m × n be the Hermite form of A and U ∈ k ( z )[ ∂ ; σ, δ ] m × m such that U A = H .(a) deg z H ij ∈ O ( m de ) for ≤ i, j ≤ n , and deg z U ij ∈ O ( m de ) for ≤ i, j ≤ m .(b) We can compute H and U with a deterministic algorithm that requires O ( nm d log( md ) · M ( m de ) + n m d · M ( m de )) or O ˜( nm d e + n m d e ) operations in k .(c) Assume k has at least m de elements. We can compute H and U with an expectednumber O ( nm d log( md ) + nm d e + n m d e ) of operations in k (using standardpolynomial arithmetic). This algorithm is probabilistic of the Las Vegas type; itnever returns an incorrect answer. Proof.
Part (a) from Theorem 5.6 (a), follows since the transformation is computedfrom an m × m submatrix of A .Part (b) follows because each iteration of the binary search requires computing theHermite form and transformation matrix U of an m × m submatrix of A . The cost ofthis is shown in Theorem 5.6. We then check if U A is in Hermite form, which is a matrixmultiplication of the m × m matrix U times the m × n matrix A . Each entry in U hasdegree O ( md ) in ∂ and degree O ( m de ) in z , so the check requires O ( nm d · M ( m de )).Both the Hermite form computation and the check are done n times, giving the totalshown cost. Note that in the event we choose an m × m submatrix which is row rankdeficient, the algorithm will fail, either in the computation of the Hermite form, or in the23nal multiplication test, and we do not need to resort to the more expensive row-reductionalgorithm outlined in Subsection 6.1.Part (c) follows similarly to part (b), except that the probabilistic algorithm describedin Theorem 5.6 (c) is used. ✷ Acknowledgement
The authors would like to thank Howard Cheng for his assistance with Section 6.1,and the anonymous referees for their exceptionally detailed and helpful reviews.They also acknowledge the support of the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) and MITACS Canada.
References
S. Abramov and M. Bronstein. On solutions of linear functional systems. In
Proc. ACMInternational Symposium on Symbolic and Algebraic Computation , pages 1–7, 2001.B. Beckermann, H. Cheng, and G. Labahn. Fraction-free row reduction of matrices ofOre polynomials.
Journal of Symbolic Computation , 41(1):513–543, 2006.Y. A. Blinkov, C. F. Cid, V. P. Gerdt, W. Plesken, and D. Robertz. The Maple packageJanet: II. linear partial differential equations. In
Proc. Workshop on Computer Algebraand Scientific Computation (CASC) , pages 41–54, 2003.M. Bronstein and M. Petkovˇsek. On Ore rings, linear operators and factorisation.
Pro-grammirovanie , 20:27–45, 1994.M. Bronstein and M. Petkovˇsek. An introduction to pseudo-linear algebra.
TheoreticalComputer Science , 157(1):3–33, 1996.D. Cantor and E. Kaltofen. Fast multiplication of polynomials over arbitrary algebras.
Acta Informatica , 28:693–701, 1991.H. Cheng.
Algorithms for Normal Forms for Matrices of Polynomialsand Ore Polynomials . PhD thesis, University of Waterloo, 2003. URL .F. Chyzak, A. Quadrat, and D. Robertz. Effective algorithms for parametrizing linearcontrol systems over Ore algebras.
Appl. Algebra Eng., Commun. Comput. , 16:319–376,2005.F. Chyzak, A. Quadrat, and D. Robertz. OreModules: A symbolic package for the studyof multidimensional linear systems.
Applications of Time Delay Systems , January2007.G. Culianez. Formes de Hermite et de Jacobson: impl´ementations et applications. Tech-nical report, INRIA, Sophia Antipolis, 2005.P. Davies, H. Cheng, and G. Labahn. Computing Popov form of general Ore polynomialmatrices. In
Milestones in Computer Algebra , pages 149–156, 2008.L.E. Dickson.
Algebras and their arithmetics . G.E. Stechert, New York, 1923.J. Dieudonn´e. Les d´eterminants sur un corps non commutatif.
Bull. Soc. Math. France ,71:27–45, 1943.J.D. Dixon. Exact solution of linear equations using p -adic expansions. Numer. Math. ,40:137–141, 1982.P. K. Draxl.
Skew Fields . Number 81 in London Mathematical Society Lecture NoteSeries. Cambridge University Press, 1983.24. von zur Gathen and J. Gerhard.
Modern Computer Algebra . Cambridge UniversityPress, Cambridge, New York, Melbourne, 2003. ISBN 0521826462.I. Gelfand, S. Gelfand, V. Retakh, and R. Wilson. Quasideterminants.
Advances inMathematics , 193(1):56–141, 2005.I. M. Gel’fand and V. S. Retakh. Determinants of matrices over non-commutative rings.
Functional Analysis and Its Applications , pages 91–102, 1991.I. M. Gel’fand and V. S. Retakh. A theory of noncommutative determinants and charac-teristic functions of graphs.
Functional Analysis and Its Applications , pages 231–246,1992.M. Giesbrecht.
Nearly Optimal Algorithms for Canonical Matrix Forms . PhD thesis,University of Toronto, 1993. 196 pp.M. Giesbrecht and A. Heinle. A polynomial-time algorithm for the Jacobson form ofa matrix of Ore polynomials. In
Proc. Computer Algebra in Scientific Computation(CASC 2012) , 2012. To appear.M. Giesbrecht and M. Kim. On computing the Hermite form of a matrix of differentialpolynomials. In
Proc. Workshop on Computer Algebra and Scientific Computation(CASC 2009) , volume 5743 of
Lecture Notes in Computer Science , pages 118–129,2009. doi: 10.1007/978-3-642-04103-7 12.M. Hal´as. An algebraic framework generalizing the concept of transfer functions tononlinear systems.
Automatica J. IFAC , 44(5):1181–1190, 2008.C. Hermite. Sur l’introduction des variables continues dans la th´eorie des nombres.
Journal f¨ur die reine und angewandte Mathematik , 41:191–216, 1851.N. Jacobson.
The Theory of Rings . American Math. Soc., New York, 1943.E. Kaltofen, M. S. Krishnamoorthy, and B. D. Saunders. Fast parallel computation ofHermite and Smith forms of polynomial matrices.
SIAM Journal of Algebraic andDiscrete Methods , 8:683–690, 1987.R. Kannan. Polynomial-time algorithms for solving systems of linear equations overpolynomials.
Theoretical Computer Science , 39:69–88, 1985.¨U. Kotta, A. Leiback, and M. Hal´as. Non-commutative determinants in nonlinear controltheory: Preliminary ideas. , pages 815–820, 2008.V. Levandovskyy and K. Schindelar. Computing diagonal form and Jacobson normalform of a matrix using Gr¨obner bases.
Journal of Symbolic Computation , 46(5):595 –608, 2011.J. Middeke. A polynomial-time algorithm for the Jacobson form for matrices of differen-tial operators. Technical Report 08-13, Research Institute for Symbolic Computation(RISC), Linz, Austria, 2008.T. Mulders and A. Storjohann. On lattice reduction for polynomial matrices.
Journal ofSymbolic Computation , 35(4):377–401, 2003.M. Newman.
Integral Matrices . Academic Press, New York, 1972.O. Ore. Linear equations in non-commutative fields.
The Annals of Mathematics , 32(3):463–477, 1931.O. Ore. Theory of non-commutative polynomials.
Annals of Mathematics , 34:480–508,1933.C. Pernet and W. Stein. Fast computation of Hermite normal forms of random integermatrices.
Journal of Number Theory , 130:1675–1683, 2010.25. Popov. Invariant description of linear, time-invariant controllable systems.
SIAM J.Control , 10:252–264, 1972.J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities.
J.Assoc. Computing Machinery , 27:701–717, 1980.H. J. S. Smith. On systems of linear indeterminate equations and congruences.
Philos.Trans. Royal Soc. London , 151:293–326, 1861.A. Storjohann. Computation of Hermite and Smith normal forms of matrices. Master’sthesis, University of Waterloo, 1994.Arne Storjohann. High-order lifting and integrality certification.
J. Symb. Comput. , 36(3-4):613–648, 2003.G. Villard. Generalized subresultants for computing the Smith normal form of polynomialmatrices.
Journal of Symbolic Computation , 20:269–286, 1995.J.H.M. Wedderburn. Non-commutative domains of integrity.
Journal f¨ur die reine undangewandte Mathematik , 167:129–141, 1932.E. Zerz. An algebraic analysis approach to linear time-varying systems.