An informal introduction to perturbations of matrices determined up to similarity or congruence
aa r X i v : . [ m a t h . R A ] D ec An informal introduction to perturbations ofmatrices determined up to similarity orcongruence
Lena Klimenko ∗ Vladimir V. Sergeichuk † Abstract
The reductions of a square complex matrix A to its canonical formsunder transformations of similarity, congruence, or *congruence areunstable operations: these canonical forms and reduction transforma-tions depend discontinuously on the entries of A . We survey resultsabout their behavior under perturbations of A and about normal formsof all matrices A + E in a neighborhood of A with respect to similarity,congruence, or *congruence. These normal forms are called miniversaldeformations of A ; they are not uniquely determined by A + E , butthey are simple and depend continuously on the entries of E . AMS classification:
Keywords: similarity, congruence, *congruence, perturbations,miniversal deformations, closure graphs.
The purpose of this survey is to give an informal introduction into the theoryof perturbations of a square complex matrix A determined up to transfor-mations of similarity S − AS , or congruence S T AS , or *congruence S ∗ AS , inwhich S is nonsingular and S ∗ ∶= ¯ S T . ∗ National Technical University of Ukraine “Kyiv Polytechnic Institute”, Prospect Per-emogy 37, Kiev, Ukraine. Email: [email protected] † Institute of Mathematics, Tereshchenkivska 3, Kiev, Ukraine. Supported by theFAPESP grant 2012/18139-2. The work was done while this author was visitingthe University of S˜ao Paulo, whose hospitality is gratefully acknowledged. Email:[email protected] J ( λ )⊕ J ( λ ) (we denote by J n ( λ ) the n × n upper-triangular Jordan block witheigenvalue λ ) is reduced by arbitrarily small perturbations to matrices ⎡⎢⎢⎢⎢⎢⎢⎢⎣ λ λ ελ λ ⎤⎥⎥⎥⎥⎥⎥⎥⎦ or ⎡⎢⎢⎢⎢⎢⎢⎢⎣ λ λ ελ λ ⎤⎥⎥⎥⎥⎥⎥⎥⎦ , ε ≠ , (1)whose Jordan canonical forms are J ( λ )⊕ J ( λ ) or J ( λ ) , respectively. There-fore, if the entries of a matrix are known only approximately, then it is unwiseto reduce it to its Jordan form.Furthermore, when investigating a family of matrices close to a givenmatrix, then although each individual matrix can be reduced to its Jordanform, it is unwise to do so since in such an operation the smoothness relativeto the entries is lost.Let J be a Jordan matrix.(a) Arnold [1] (see also [2, 3]) constructed a miniversal deformation of J ;i.e., a simple normal form to which all matrices J + E close to J canbe reduced by similarity transformations that smoothly depend on theentries of E .(b) Boer and Thijsse [6] and, independently, Markus and Parilis [22] foundeach Jordan matrix J ′ for which there exists an arbitrary small matrix E such that J + E is similar to J ′ . For example, if J = J ( λ ) ⊕ J ( λ ) ,then J ′ is either J , or J ( λ ) ⊕ J ( λ ) , or J ( λ ) with the same λ (see(1)).Using (b), it is easy to construct for small n the closure graph G n forsimilarity classes of n × n complex matrices; i.e., the Hasse diagram of thepartially ordered set of similarity classes of n × n matrices that are orderedas follows: a ≼ b if a is contained in the closure of b . Thus, the graph G n shows how the similarity classes relate to each other in the affine space of n × n matrices.In Section 2.1 we give a sketch of constructive proof of Arnold’s theoremabout miniversal deformations of Jordan matrices, and in Sections 2.2–2.42e consider closure graphs for similarity classes and similarity bundles. InSections 3 and 4 we survey analogous results about perturbations of matricesdetermined up to congruence or *congruence.We do not survey the well-developed theory of perturbations of matrixpencils [9, 10, 11, 15, 18, 19]; i.e., of matrix pairs ( A, B ) up to equivalencetransformations ( RAS, RBS ) with nonsingular R and S .All matrices that we consider are complex matrices. In this section, we formulate Arnold’s theorem about miniversal deforma-tions of matrices under similarity and give a sketch of its constructive proof.Since each square matrix is similar to a Jordan matrix, it suffices to studyperturbations of Jordan matrices.For each Jordan matrix J = t ⊕ i = ( J m i ( λ i ) ⊕ ⋅ ⋅ ⋅ ⊕ J m iri ( λ i )) , m i ⩾ m i ⩾ . . . ⩾ m ir i (2)with λ i ≠ λ j if i ≠ j , we define the matrix of the same size J + D ∶ = t ⊕ i = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ J m i ( λ i ) + ↓ ↓ . . . ↓ ← J m i ( λ i ) + ↓ ⋅ ⋅ ⋅ ⋮⋮ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ↓ ← . . . ← J m iri ( λ i ) + ↓ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (3)in which 0 ← ∶ = ⎡⎢⎢⎢⎢⎢⎣ ∗ . . . ⋮ ⋮ ⋮∗ . . . ⎤⎥⎥⎥⎥⎥⎦ and 0 ↓ ∶ = ⎡⎢⎢⎢⎢⎢⎢⎢⎣ ⋯ ⋮ ⋮ ⋯ ∗ ⋯ ∗ ⎤⎥⎥⎥⎥⎥⎥⎥⎦ are blocks whose entries are zeros and stars.3he following theorem of Arnold [1, Theorem 4.4] is also given in [2,Section 3.3] and [3, § Theorem 2.1 ([1]) . Let J be the Jordan matrix (2) . Then all matrices J + X that are sufficiently close to J can be simultaneously reduced by sometransformation J + X ↦ S ( X ) − ( J + X )S ( X ) , S ( X ) is analyticat and S ( ) = I, (4) to the form J + D defined in (3) whose stars are replaced by complex numbersthat depend analytically on the entries of X . The number of stars is minimalthat can be achieved by transformations of the form (4) , it is equal to thecodimension of the similarity class of J . The matrix (3) with independent parameters instead of stars is called a miniversal deformation of J (see formal definitions in [1], [2], or [3]).The codimension of the similarity class of J is defined as follows. Foreach A ∈ C n × n and a small matrix X ∈ C n × n , ( I − X ) − A ( I − X ) = ( I + X + X + ⋯ ) A ( I − X )= A + ( XA − AX ) + X ( XA − AX ) + X ( XA − AX ) + ⋯ = A + XA − AX ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ small + X ( I − X ) − ( XA − AX )´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ very small and so the similarity class of A in a small neighborhood of A can be obtainedby a very small deformation of the affine matrix space { A + XA − AX ∣ X ∈ C n × n } . (By the Lipschitz property [24], if A and B are close to each otherand B = S − AS with a nonsingular S , then S can be taken near I n .)The vector space T ( A ) ∶ = { XA − AX ∣ X ∈ C n × n } is the tangent space to the similarity class of A at the point A . The numbersdim C T ( A ) , codim C T ( A ) ∶ = n − dim C T ( A ) (5)are called the dimension and codimension of the similarity class of A .4 emark . The matrix (3) is the direct sum of t matrices that are not blocktriangular. But each Jordan matrix J is permutation similar to some Weyrmatrix J with the following remarkable property: all commuting with J matrices are upper block triangular. Producing with (3) the same transfor-mations of permutation similarity, Klimenko and Sergeichuk [19] obtained anupper block triangular matrix J + D , which is a miniversal deformation of J .Now we show sketchily how all matrices near J can be reduced to the form(3) by near-identity elementary similarity transformations; which explainsthe structure of the matrix (3). Lemma 2.1.
Two matrices are similar if and only if one can be transformedto the other by a sequence of the following transformations ( which are called elementary similarity transformations ; see [25, Section 1.40] ) : (i) Multiplying column i by a nonzero a ∈ C ; then dividing row i by a . (ii) Adding column i multiplied by b ∈ C to column j ; then subtracting row j multiplied by b from row i . (iii) Interchanging columns i and j ; then interchanging rows i and j .Proof. Let A and B be similar; that is, S − AS = B . Write S as a product ofelementary matrices: S = E E ⋯ E t . Then A ↦ E − AE ↦ E − E − AE E ↦ ⋅ ⋅ ⋅ ↦ E − t ⋯ E − E − AE E ⋯ E t = B is a desired sequence of elementary similarity transformations. Sketch of the proof of Theorem 2.1.
Two cases are possible.
Case 1: t =
1. Suppose first that J = J ( ) ⊕ J ( ) . Let J + E = [ b ij ] i,j = ∶ = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ε + ε ε ε ε ε ε + ε ε ε ε ε ε ε ε ε ε ε ε + ε ε ε ε ε ε ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (6)5e any matrix near J (i.e., all ε ij are small). We need to reduce it to theform ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , (7)in which the ∗ ’s are small complex numbers, by those transformations fromLemma 2.1 that are close to the identity transformation.Dividing column 2 of (6) by 1 + ε and multiplying row 2 by 1 + ε (transformation (i)), we make b =
1. Since ε is small, this transformationis near-identity and the obtained matrix is near J . Some b ij and ε ij havebeen changed, but we use the same notation for them.Subtracting column 2 (with ε =
0) multiplied by ε from column 1, wemake b =
0; the inverse transformation of rows (which must be done bythe definition of transformation (ii)) slightly changes row 2. Analogously, wemake b = b = b = [ b ij ] i,j = = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ε ε + ε ε ε ε ε ε ε ε ε ε ε ε + ε ε ε ε ε ε ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ with row 1 as in (7). In the same manner, we make b = + ε , and then b = b = b = b = b =
1; then b = b = b = b = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ( ∗ ’s are small numbers )
6y using near-identity elementary similarity transformations with (6).To reduce the number of stars, we subtract row 2 multiplied by b fromrow 5 making b =
0. The inverse transformation of columns adds column 5multiplied by the old b to column 2. Then we make b = b = b , b , and b .We have simultaneously reduced all matrices (6) near J to the form (7)by a similarity transformation that analytically depends on all ε ij and thatis identity if all ε ij = J ( ) + E near a nilpotent Jordan matrix J ( ) ∶ = J m ( ) ⊕ ⋅ ⋅ ⋅ ⊕ J m r ( ) , m ⩾ m ⩾ . . . ⩾ m r can be reduced first to matrices of the form ⎡⎢⎢⎢⎢⎢⎣ J m ( ) + ↓ . . . ↓ ⋮ ⋅ ⋅ ⋅ ⋮ ↓ . . . J m r ( ) + ↓ ⎤⎥⎥⎥⎥⎥⎦ and then to matrices of the form (3) with t = λ =
0, and m , . . . , m r instead of m , . . . , m r .This proves the theorem for each Jordan matrix J ( λ ) = J ( ) + λI with asingle eigenvalue λ since S ( E ) − ( J ( λ ) + E ) S ( E ) = S ( E ) − ( J ( ) + E ) S ( E ) + λI . Case 2: t ⩾
2. In this case, (2) has distinct eigenvalues. Write (2) in the form J = J ⊕ ⋅ ⋅ ⋅ ⊕ J t , where each J i ∶ = J m i ( λ i ) ⊕ ⋅ ⋅ ⋅ ⊕ J m iri ( λ i ) is of size n i × n i and has the single eigenvalue λ i . Let J + E = ⎡⎢⎢⎢⎢⎢⎣ J + E . . . E t ⋮ ⋅ ⋅ ⋅ ⋮ E t . . . J t + E tt ⎤⎥⎥⎥⎥⎥⎦ (8)be any matrix near J (i.e., all E ij are small). We make E ij = i ≠ j by near-identity similarity transformations as follows.Represent (8) in the form J + E ⇙ + E ⇗ in which J + E ⇙ ∶ = ⎡⎢⎢⎢⎢⎢⎢⎢⎣ J E J ⋮ ⋱ ⋱ E t . . . E t,t − J t ⎤⎥⎥⎥⎥⎥⎥⎥⎦ , E ⇗ ∶ = ⎡⎢⎢⎢⎢⎢⎢⎢⎣ E E . . . E t E ⋱ ⋮⋱ E t − ,t E tt ⎤⎥⎥⎥⎥⎥⎥⎥⎦ . J + E ⇙ . Add to its first vertical strip the second stripmultiplied by any n × n matrix M to the right. Make the inverse trans-formation of rows: subtract from the second horizontal strip the first stripmultiplied by M to the left. This similarity transformation replaces E with E + J M − M J . Since J and J have distinct eigenvalues, there exists M for which E + J M − M J = § M issmall since E is small.In the same manner, we successively make zero the other blocks of thefirst underdiagonal E , E , . . . , E t,t − of J + E ⇙ , then the blocks of its secondunderdiagonal E , . . . , E t,t − , and so on. Thus, there exists a near-identitymatrix S such that S − ( J + E ⇙ ) S = J ⊕ ⋅ ⋅ ⋅ ⊕ J t .We make the same similarity transformation with the whole matrix J + E = J + E ⇙ + E ⇗ and obtain the matrix J + E ′ ∶ = S − ( J + E ) S . Its underdiagonalblocks E ′ ij ( i > j ) coincide with the underdiagonal blocks of S − E ⇗ S , whichare very small since all E ij are small and the transformation is near-identity.We apply the same reduction to J + E ′ and obtain a matrix J + E ′′ = S − ( J + E ′ ) S whose underdiagonal blocks E ′′ ij ( i > j ) are very very small,and so on.The infinite product S S . . . converges to a near-identity matrix S suchthat all underdiagonal blocks of J + ˜ E ∶ = S − ( J + E ) S are zero.By near-identity similarity transformations, we successively make zero thefirst overdiagonal ˜ E , ˜ E , . . . , ˜ E t − ,t of J + ˜ E , then its second overdiagonal˜ E , . . . , ˜ E t − ,t , and so on.We have reduced (8) to the block diagonal form ( J + F ) ⊕ ⋅ ⋅ ⋅ ⊕ ( J t + F t ) in which all F i are small. Reducing each summand J i + F i as in Case 1, weobtain a matrix of the form (3). Remark . In the above proof we have described sketchily how to constructthe transformation (4). Algorithms for constructing this transformation arediscussed in [20, 21].
Let J be a Jordan matrix and let λ be its eigenvalue. Denote by w λj thenumber of Jordan blocks J m ( λ ) of size m ⩾ j in J ; the sequence ( w λ , w λ , . . . ) is called the Weyr characteristic of J (and of any matrix that is similar to J ) for the eigenvalue λ . 8he following theorem was proved by Boer and Thijsse [6] and, inde-pendently, by Markus and Parilis [22]; another proof was given by Elmroth,Johansson, and K˚agstr¨om [10, Theorem 2.2]. Theorem 2.2 ([6, 22]) . Let J and J ′ be Jordan matrices of the same size.Then J can be transformed to a matrix that is similar to J ′ by an arbitrarilysmall perturbation if and only if J and J ′ have the same set of eigenvalueswith the same multiplicities, and their Weyr characteristics satisfy w λ ⩾ w ′ λ , w λ + w λ ⩾ w ′ λ + w ′ λ , w λ + w λ + w λ ⩾ w ′ λ + w ′ λ + w ′ λ , . . . for each eigenvalue λ . Theorem 2.2 was extended to Kronecker’s canonical forms of matrix pen-cils by Pokrzywa [23].
Definition 2.1.
Let T be a topological space with an equivalence relation.The closure graph (or closure diagram ) is the directed graph whose verticesbijectively correspond to the equivalence classes and for equivalence classes a and b there is a directed path from a vertex of a to a vertex of b if and onlyif a ⊂ b , in which b denotes the closure of b .Thus, the closure graph is the Hasse diagram of the set of equivalenceclasses with the following partial order: a ≼ b if and only if a ⊂ b . The closuregraph shows how the equivalence classes relate to each other in T .In this section, T = C n × n and the equivalence relation is the similarityof matrices. Since each similarity class contains exactly one Jordan matrixdetermined up to permutations of Jordan blocks, we identify the verticeswith the Jordan matrices determined up to permutations of Jordan blocks.Theorem 2.2 admits to construct the closure graphs due to the followinglemma. Lemma 2.2.
The closure graph for similarity classes of n × n matrices con-tains a directed path from a Jordan matrix J to a Jordan matrix J ′ if andonly if J can be transformed to a matrix that is similar to J ′ by an arbitrarilysmall perturbation. roof. Denote by [ M ] the similarity class of a square matrix M .“ ⇐Ô ” Let J can be transformed to a matrix that is similar to J ′ byan arbitrarily small perturbation. Then there exists a sequence of matrices J + E , J + E , J + E , . . . in [ J ′ ] that converges to J . This means that J ∈ [ J ′ ] . Let A ∈ [ J ] ; i.e., A = S − J S for some S . Then the sequence ofmatrices S − ( J + E i ) S = A + S − E i S ( i = , , . . . ) in [ J ′ ] converges to A , andso A ∈ [ J ′ ] . Therefore, [ J ] ⊂ [ J ′ ] and there is a directed path from J to J ′ . Corollary 2.1.
By Theorem 2.2, the arrows are only between Jordan matri-ces with the same sets of eigenvalues. Let J be a Jordan matrix. • Let J ′ be a Jordan matrix of the same size. Each neighborhood of J contains a matrix whose Jordan canonical form is J ′ if and only if thereis a directed path from J to J ′ ( if J = J ′ then there always exists the“lazy” path of length from J to J ′ ) . • The closure of the similarity class of J is equal to the union of thesimilarity classes of all Jordan matrices J ′ such that there is a directedpath from J ′ to J ( if J = J ′ then there always exists the “lazy” path ) .Example . Let us construct the closure graph for similarity classes of4 × J m ( λ ) .Replacing them by λ m and deleting the symbols ⊕ , we get the compactnotation of Jordan matrices which was used by Arnold [1]. For example, λ λµ is J ( λ ) ⊕ J ( λ ) ⊕ J ( µ ) (9)(we write λ, µ instead of λ , µ ).For all Jordan matrices of size 4 × ( w , w + w , matrix ristic ( w , w , w , w ) w + w + w , w + w + w + w )
00 (3,1,0,0) (3,4,4,4)0 (2,2,0,0) (2,4,4,4)0 (1,1,1,1) (1,2,3,4)10sing this table, Theorem 2.2, and Lemma 2.2, it is easy to construct thefollowing closure graph for similarity classes of nilpotent × matrices:0000 → → → → In the same way, one can construct the closure graph for similarity classesof all 4 × λ λ µ λ µ λ µν λµνξ dimension 12 λ λ O O λ λµ O O λ µµ O O λλµν O O dimension 10 λ λ O O λλµµ O O dimension 8 λ λλ O O λλλµ O O dimension 6 λλλλ O O dimension 0 (10)Figure 1: The closure graph for similarity classes of × matrices λ, µ, ν, ξ are arbitrary distinct complex numbers. The similarity classes of4 × J that are located at the same horizontal level in (10)have the same dimension (defined in (5)), which is indicated to the right andis calculated as follows: it equals 16 − codim C T ( J ) , in which codim C T ( J ) isthe number of stars in (3) (see (5) and Theorem 2.1). For example, if J is(9) with λ ≠ µ , then (3) is ⎡⎢⎢⎢⎢⎢⎢⎢⎣ λ ∗ λ + ∗ ∗ ∗ λ + ∗
00 0 0 µ + ∗ ⎤⎥⎥⎥⎥⎥⎥⎥⎦ and so dim C ( J ) = − codim C T ( J ) = − = . The following example shows that the structure of the closure graph forlarger matrices is not so simple as in (10).
Example . The closure graph for similarity classes of 6 × dim 300 O O dim 280 O O dim 260 ♠♠♠♠♠ h h PPPPP dim 240 h h ◗◗◗◗ ♥♥♥♥ dim 220 ♠♠♠♠ h h PPPP dim 180 h h ◗◗◗◗ ♥♥♥ dim 160 O O dim 10000000 O O dim 0Figure 2: The closure graph for similarity classes of × nilpotent matrices Arnold [1, § bundle of matrices under similarity as a set of allmatrices having the same Jordan type , which is defined as follows: matrices A and B have the same Jordan type if there is a bijection from the set of distincteigenvalues of A to the set of distinct eigenvalues of B that transforms theJordan canonical form of A to the Jordan canonical form of B . For example,the Jordan matrices J ( ) ⊕ J ( ) ⊕ J ( ) , J ( ) ⊕ J ( ) ⊕ J ( − ) belong to the same bungle. All matrices of a bundle have similar propertiesand not only with respect to perturbations; for example, its Jordan canonicalmatrices have the same set of commuting matrices.12ote that the closure graph for bundles of n × n matrices under similarityhas a finite number of vertices; moreover, it is in some sense more informativethan the closure graph for similarity classes. For example, one cannot seefrom the latter graph that each neighborhood of J n ( λ ) contains a matrixwith n distinct eigenvalues (since there is no diagonal matrix whose similarityclass has a nonzero intersection with each neighborhood of J n ( λ ) ). But theclosure graph for bundles has an arrow from the bundle containing J n ( λ ) tothe bundle of all matrices with n distinct eigenvalues.Furthermore, not every convergent sequence of n × n matrices B , B , . . . → A, (11)in which all B i are not similar to A , gives a directed path in the closure graphfor similarity classes. But every sequence (11), in which all B i do not belongto the bundle A that contains A , gives at least one directed path in theclosure graph for similarity bundles. Indeed, the number of bundles of n × n matrices is finite, and so there is an infinite subsequence B n , B n , . . . → A in which all B n i belong to the same bundle B . Hence A ∈ B . One can provethat A ⊂ B . Example . The closure graph for similarity bundles of 4 × λλλλ → λ λλ → ⋯ → λ , λλλµ → λ λµ → λ µ, . . . , λµνξ (13)by replacing their parameters by unequal complex numbers (the numbersof parameters in the vertices of the linear subgraphs (13) are equal to 1,2, 2, 3, 4, respectively). Thus, although the sequences of Greek letters inthe vertices of (10) and (12) are the same, each vertex of (10) representsan infinite set of similarity classes whose matrices have the same Jordantype (and so these similarity classes have the same dimension), whereas thecorresponding vertex in (12) represents only one bundle, which is the unionof these similarity classes; its dimension is equal to the dimension of any ofits similarity classes plus the number of parameters. Notice that each arrowof (10) corresponds to an arrow of (12), but (12) has additional arrows.13 µνξ dim 16 λ µν ♥♥♥ dim 15 λ µ ❤❤❤❤❤❤❤❤❤❤ λ µ ♣♣ dim 14 λ qqqq ❤❤❤❤❤❤❤❤❤❤❤ λλµν O O dim 13 λ λµ O O ❤❤❤❤❤❤❤❤❤ λ µµ O O ♣♣ dim 12 λ λ O O ♣♣ ❣❣❣❣❣❣❣❣❣❣ dim 11 λλµµ O O dim 10 λ λ O O ❣❣❣❣❣❣❣❣❣ dim 9 λλλµ O O dim 8 λ λλ ♦♦ O O dim 7 λλλλ O O dim 1 (12)Figure 3: The closure graph for similarity bundles of × matrices Dmytryshyn, Futorny, and Sergeichuk [7] constructed miniversal deforma-tions of the following congruence canonical matrices given by Horn and Serge-ichuk [16, 17]:
Every square complex matrix is congruent to a direct sum, deter-mined uniquely up to permutation of summands, of matrices ofthe form [ I m J m ( λ ) ] , ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⋰− ⋰ − −
11 1 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , J k ( ) , in which λ ∈ C ∖ { , ( − ) m + } and is determined up to replacementby λ − . × × Theorem 3.1 ([7, Example 2.1]) . Let A be any × or × matrix. Then allmatrices A + X that are sufficiently close to A can be simultaneously reducedby some transformation S ( X ) T ( A + X ) S ( X ) , S ( X ) is holomorphic at , (14) to one of the following forms, in which λ ∈ C ∖ { − , } and each nonzero λ isdetermined up to replacement by λ − . ● If A is × : [ ] + [ ∗ ∗∗ ∗ ] , [ ] + [ ∗ ∗ ] , [ ] + [ ∗ ] , [ − ] + [ ∗ ∗ ∗ ] , [ −
11 1 ] + [ ∗
00 0 ] , [ λ ] + [ ∗ ] . ● If A is × : ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ − ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ λ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ ( λ ≠ ) , ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ −
11 1 0 ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ − ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗
00 0 0 ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ λ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ −
11 1 1 ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎢⎢⎢⎢⎢⎣ − −
11 1 0 ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ⎤⎥⎥⎥⎥⎥⎦ . Each of these matrices has the form A can + D in which A can is a canonicalmatrix for congruence and the stars in D are complex numbers that tend tozero as X tends to zero. The number of stars is the smallest that can beattained by using transformations (14) ; it is equal to the codimension of thecongruence class of A . The codimension of the congruence class of a congruence canonical matrix A ∈ C n × n was calculated by Dmytryshyn, Futorny, and Sergeichuk [7] andindependently by De Ter´an and Dopico [4]; it is defined as follows. For eachsmall matrix X ∈ C n × n , ( I + X ) T A ( I + X ) = A + X T A + AX ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ small + X T AX ´¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¶ very small and so the congruence class of A in a small neighborhood of A can be obtainedby a very small deformation of the affine matrix space { A + X T A + AX ∣ X ∈ C n × n } . (By the local Lipschitz property [24], if A and B are close to eachother and B = S T AS with a nonsingular S , then S can be taken near I n .)The vector space T ( A ) ∶ = { X T A + AX ∣ X ∈ C n × n } is the tangent space to the congruence class of A at the point A . The numbersdim C T ( A ) , codim C T ( A ) ∶ = n − dim C T ( A ) are called the dimension and codimension of the congruence class of A .Congruence bundles are defined by Futorny, Klimenko, and Sergeichuk[12] via bundles of matrix pairs under equivalence. Recall, that pairs ( A, B ) and ( A ′ , B ′ ) of m × n matrices are equivalent if there are nonsingular R and S such that RAS = A ′ and RBS = B ′ . By Kronecker’s theorem about matrixpencils [14, Chapter XII, § ( A, B ) of matrices of the same sizeis equivalent to L ⊕ P ( λ ) ⊕ ⋅ ⋅ ⋅ ⊕ P t ( λ t ) , λ i ≠ λ j if i ≠ j, λ , . . . , λ t ∈ C ∪ ∞ , (15)16n which L is a direct sum of pairs of the form ( L k , R k ) and ( L Tk , R Tk ) , k = , , . . . , defined by L k ∶ = ⎡⎢⎢⎢⎢⎢⎣ ⋱ ⋱ ⎤⎥⎥⎥⎥⎥⎦ , R k ∶ = ⎡⎢⎢⎢⎢⎢⎣ ⋱ ⋱ ⎤⎥⎥⎥⎥⎥⎦ ( ( k − ) -by- k ) , and each P i ( λ i ) is a direct sum of pairs of the form ( I k , J k ( λ i )) if λ i ∈ C or ( J k ( ) , I k ) if λ i = ∞ .The direct sums L and P i ( λ i ) are determined by ( A, B ) uniquely, up topermutation of summands. The equivalence bundle of (15) consists of allmatrix pairs that are equivalent to pairs of the form L ⊕ P ( µ ) ⊕ ⋅ ⋅ ⋅ ⊕ P t ( µ t ) , µ i ≠ µ j if i ≠ j, µ , . . . , µ t ∈ C ∪ ∞ , with the same L , P , . . . , P t (see [9]).The definition of bundles of matrices under congruence is not so evident.They could be defined via the congruence canonical form by analogy withbundles of matrices under similarity and bundles of matrix pairs, but, unlikethe Jordan and Kronecker canonical forms, the perturbation behavior of acongruence canonical matrix with parameters depends on the values of itsparameters , which is illustrated by the canonical matrices [ − ] and [ λ ] in the left graph in Figure 4. Definition 3.1 ([12]) . Two square matrices A and B are in the same con-gruence bundle if and only if the pairs ( A, A T ) and ( B, B T ) are in the sameequivalence bundle.Definition 3.1 is based on Roiter’s statement (see [12, Lemma 4.1]): two n × n matrices A and B are congruent if and only if the pairs ( A, A T ) and ( B, B T ) are equivalent. Example . The closure graphs for congruence classes and congruence bun-dles of 2 × The left graph in Figure 4 is the closure graph for congruence classes of2 × × [ λ ] represents theinfinite set of vertices indexed by λ ∈ C ∖ { − , } .17 −
11 1 ] [ λ ] [ ][ ] O O _ _ ❄❄❄❄ @ @ ✁✁✁✁ [ − ] O O [ ] b b ❊❊ O O {[ λ ]} λ dim 4 [ −
11 1 ] O O [ ] ` ` ❇❇❇❇❇ dim 3 [ ] O O ` ` ❇❇❇❇❇ dim 2 [ − ] O O dim 1 [ ] d d ❏❏❏❏ O O dim 0Figure 4: The closure graphs for congruence classes and congruence bundles of × matrices, in which λ ∈ C ∖ {− , } and each nonzero λ is determined up toreplacement by λ − . The right graph is the closure graph for congruence bundles of 2 × {[ λ ]} λ represents the bundle that consists of allmatrices whose congruence canonical forms are [ λ ] with λ ≠ ±
1. Theother vertices are canonical matrices; their bundles coincide with theircongruence classes. Note that [ − ] and [ λ ] ( λ ≠ ±
1) properly belongto distinct bundles because these matrices have distinct properties withrespect to perturbations, which is illustrated by the left graph. Otherarguments in favor of Definition 3.1 of congruence bundles are given in[12, Section 6].The congruence classes and bundles with vertices on the same horizontallevel have the same dimension, which is indicated to the right.
Example . The closure graphs for congruence classes and congruence bun-dles of 3 × The left graph in Figure 5 is the closure graph for congruence classes of3 × × −
11 1 1 ] [ µ ] [ − −
11 1 0 ][ ] ; ; ①①①①① c c ●●●●● O O [ − ] O O [ ] O O [ −
11 1 0 ] D D ✠✠✠✠✠✠✠✠✠✠✠✠✠✠ O O [ λ ] O O [ ] Z Z ✹✹✹✹✹✹✹✹✹✹✹✹✹ O O [ − ] O O [ ] O O c c ●●●●● ; ; ✇✇✇✇✇ [ ] c c ●●●●● O O {[ µ ]} µ dim 9 [ −
11 1 1 ] : : ✉✉✉✉✉ [ − −
11 1 0 ] d d ❏❏❏❏❏ dim 8 [ ] tttttt d d ■■■■■■ dim 7 [ − ] O O {[ λ ]} λ O O [ ] O O dim 6 [ −
11 1 0 ] : : ✉✉✉✉✉ O O [ ] e e ❏❏❏❏❏❏ O O dim 5 [ − ] O O [ ] d d ■■■■■■ tttttt dim 3 [ ] d d ■■■■■■ O O dim 0Figure 5: The closure graphs for congruence classes and congruence bundles of × matrices, in which λ, µ ≠ ± , and nonzero λ and µ are determined up toreplacements by λ − and µ − . matrices for congruence. The graph is infinite: [ λ ] and [ µ ] represent the infinite sets of vertices indexed by λ, µ ≠ ± The right graph is the closure graph for congruence bundles of 3 × {[ λ ]} λ and {[ µ ]} µ represent the bundles thatconsist of all matrices whose congruence canonical forms are [ λ ]( λ ≠ ±
1) or [ µ ] ( µ ≠ ± Remark . Let M be a 2 × × • Let N be another canonical matrix for congruence of the same size.Each neighborhood of M contains a matrix from the congruence class19respectively, bundle) of N if and only if there is a directed path from M to N in the left (resp. right) graph in Figures 4 or 5. Note thatthere always exists the “lazy” path of length 0 from M to M if M = N . • The closure of the congruence class (resp. bundle) of M is equal tothe union of the congruence classes (resp. bundles) of all canonicalmatrices N such that there is a directed path from N to M . Dmytryshyn, Futorny, and Sergeichuk [8] constructed miniversal deforma-tions of the following *congruence canonical matrices given by Horn andSergeichuk [16, 17]:
Every square complex matrix is *congruent to a direct sum, de-termined uniquely up to permutation of summands, of matricesof the form [ I m J m ( λ ) ] , µ ⎡⎢⎢⎢⎢⎢⎢⎢⎣ ⋰ i ⋰ i ⎤⎥⎥⎥⎥⎥⎥⎥⎦ , J k ( ) , (16) in which λ, µ ∈ C , ∣ λ ∣ > , and ∣ µ ∣ = . (The condition ∣ λ ∣ > < ∣ λ ∣ < × × Theorem 4.1.
Let A be any × or × matrix. Then all matrices A + X that are sufficiently close to A can be simultaneously reduced by sometransformation S ( X ) ∗ ( A + X ) S ( X ) , S ( X ) is nonsingular and conti-nuous on a neighborhood of zero,to one of the following forms. If A is × : [ ] + [ ∗ ∗∗ ∗ ] , [ µ
00 0 ] + [ ε ∗ ∗ ] , [ µ µ ] + [ ε δ ε ] , [ µ µ iµ ] + [ ∗
00 0 ] , [ λ ] + [ ∗ ] . ● If A is × : ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ µ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ε ∗ ∗ ∗∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ µ µ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ε δ ε ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ µ µ µ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ε δ ε δ δ ε ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ µ µ iµ µ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ δ ε ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ µ µ iµ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ λ µ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ε ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ λ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ ( λ ≠ ) , ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ∗ ∗ ⎤⎥⎥⎥⎥⎥⎦ , ⎡⎢⎢⎢⎢⎢⎣ µ µ iµ µ iµ ⎤⎥⎥⎥⎥⎥⎦ + ⎡⎢⎢⎢⎢⎢⎣ ε
00 0 0 ⎤⎥⎥⎥⎥⎥⎦ , Each of these matrices has the form A can + D , in which A can is a canonicalmatrix for *congruence, the stars in D are complex numbers, ∣ λ ∣ < , ∣ µ ∣ = ∣ µ ∣ = ∣ µ ∣ = , and ε l ∈ R if µ l ∉ R δ lr = if µ l ≠ ± µ r ε l ∈ i R if µ l ∈ R δ lr ∈ C if µ l = ± µ r ( Clearly, D tends to zero as X tends to zero. ) For each A can + D , twicethe number of its stars plus the number of its entries ε l , δ lr is equal to thecodimension over R of the *congruence class of A can . A ∈ C n × n was calculated by De Ter´an and Dopico [5] and independentlyby Dmytryshyn, Futorny, and Sergeichuk [8]; it is defined as follows. Foreach A ∈ C n × n and a small matrix X ∈ C n × n , ( I + X ) ∗ A ( I + X ) = A + X ∗ A + AX ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ small + X ∗ AX ´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¶ very small and so the *congruence class of A in a small neighborhood of A can beobtained by a very small deformation of the real affine matrix space { A + X ∗ A + AX ∣ X ∈ C n × n } . (By the local Lipschitz property [24], if A and B areclose to each other and B = S ∗ AS with a nonsingular S , then S can be takennear I n ). The real vector space T ( A ) ∶ = { X ∗ A + AX ∣ X ∈ C n × n } is the tangent space to the *congruence class of A at the point A . Thenumbers dim R T ( A ) , codim R T ( A ) ∶ = n − dim R T ( A ) are called the dimension and, respectively, codimension over R of the *con-gruence class of A . Example . The closure graph for *congruence classes of 2 × [ ] represents an infinite set of vertices indexed by the parametersof the corresponding canonical matrix. The *congruence classes of canonicalmatrices that are located at the same horizontal level in (17) have the samedimension over R , which is indicated to the right. The arrow [ λ
00 0 ] → [ µ ν ] exists if and only if λ = µa + νb for some nonnegative a, b ∈ R . The arrow [ λ
00 0 ] → [ ττ iτ ] exists if and only if the imaginary part of λ ¯ τ is nonnega-tive. The arrow [ λ − λ ] → [ ττ iτ ] exists if and only if τ = ± λ . The arrows [ λ
00 0 ] → [ λ ± λ ] exist if and only if the value of λ is the same in both matrices.The other arrows exist for all values of parameters of their matrices. Remark . Let M be a 2 × • Let N be another 2 × M contains a matrix that is *congruent to N if and onlyif there is a directed path from M to N in (17) (if M = N , then therealways exists the “lazy” path of length 0 from M to N ).22 µ ν ] [ σ ] [ ττ τ i ] ∣ µ ∣ = ∣ ν ∣ = ∣ τ ∣ = ,µ ≠ ± ν, ∣ σ ∣ < , dim R [ λ λ ] [ λ − λ ] τ =± λ O O dim R [ λ
00 0 ] the same λ ✸✸✸✸✸ Y Y ✸✸✸✸✸ the same λ ✟✟✟✟✟ D D ✟✟✟✟✟ O O λ ∈ µ R + + ν R + ✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮ T T ✮✮✮✮✮ Im ( λ ¯ τ )⩾ ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔ J J ✔✔✔✔✔ ∣ λ ∣ = , dim R [ ] O O G G X X dim R The closure graph for *congruence classes of × matrices, in which R + denotes the set of nonnegative real numbers, Im ( c ) denotes the imaginary partof c ∈ C , and λ, µ, ν, σ, τ ∈ C . • The closure of the *congruence class of M is equal to the union of the*congruence classes of all canonical matrices N such that there is adirected path from N to M . References [1] V.I. Arnold, On matrices depending on parameters, Russian Math. Sur-veys 26 (2) (1971) 29–43.[2] V.I. Arnold, Lectures on bifurcations in versal families, Russian Math.Surveys 27 (5) (1972) 54–123.[3] V.I. Arnold, Geometrical Methods in the Theory of Ordinary DifferentialEquations, Springer-Verlag, 1988.234] F. De Ter´an, F.M. Dopico, The solution of the equation XA + AX T = XA + AX ∗ = ∼∼