STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES We obtain some further results for pairs of commuting matrices. We show that a pair of commutin

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1 On the stability of invariant subspaces of commuting matrices Tomaz Kosir and Bor Plestenjak September 18, 001 Abstract We study the stability of (joint) invariant subspaces of a nite set of commuting matrices. We generalize some of the results of Gohberg, Lancaster, and Rodman for the single matrix case. For sets of two or more commuting matrices we exhibit some phenomena dierent from the single matrix case. We show that each root subspace is a stable invariant subspace, that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable, and that, even in the derogatory case, the eigenspace is stable if it is one-dimensional. We prove that a pair of commuting matrices has only nitely many stable invariant subspaces. At the end, we discuss the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem. 1 Introduction In the paper we study the stability of invariant subspaces of k-tuples (k ) of commuting matrices. The problem of stability arose in applications to multiparameter eigenvalue problems [1]. The stability is crucial when numerical calculations are performed to nd a basis of an invariant subspace [1]. In this paper, an invariant (resp. root) subspace of a k-tuple of commuting matrices always refers to a joint invariant (resp. root) subspace of the k-tuple. In the single matrix case (i.e., if k = 1) Gohberg, Lancaster, and Rodman [] characterized all stable invariant subspaces. They showed that each root subspace is stable, each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable, and that direct sums of these two types of subspaces are the only stable invariant subspaces. We generalise most of these results. We show that each root subspace is stable, and that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable. Moreover, if there is only one invariant subspace of a root subspace of a given dimension then it is stable. In particular, if the eigenspace is one-dimensional then it is stable. We show that also direct sums of these types of subspaces are stable. However, we do not know if these are the only possible stable subspaces of a k-tuple of commuting matrices. Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia. tomaz.kosir@fmf.uni-lj.si and bor.plestenjak@fmf.uni-lj.si The research was supported in part by the Ministry of Science and Technology of Slovenia. 1

2 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES We obtain some further results for pairs of commuting matrices. We show that a pair of commuting matrices has only nitely many stable invariant subspaces. We consider a few examples and state a number of open problems. We conclude with some results on the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem. Such a problem has an associated k-tuple of commuting matrices. (See x for a brief introduction and [1] for details.) Plestenjak [1] studied a numerical algorithm for computing a basis of a root subspace at a nonderogatory eigenvalue of an associated k-tuple of commuting matrices. Since each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable there is no problem of stability in the algorithm presented in [1]. Preliminaries Let A = fa 1 ; : : : ; A k g, (k ), be a set of commuting n n matrices over C. We say that a subspace N of C n is A{invariant if A l N N ; l = 1; : : : ; k: The set of all A{invariant subspaces is denoted by Inv(A). A k-tuple = ( 1 ; : : : ; k ) C k is an eigenvalue of a set of commuting matrices A if Ker(A I) := k\ l=1 Ker(A l l I) = f0g: A nonzero vector z Ker(A I) is an eigenvector for and A. The root subspace for an eigenvalue is denoted by R (A) and is equal to \ h i Ker (A 1 1 I) l 1 (A k k I) l k l1++l k =n An eigenvalue is called geometrically simple if dim Ker(A I) = 1. nonderogatory if dim k\ l=1 It is called Ker(A l l I) j = j; (1) for j = 1; ; : : : ; dim R (A). We say that an eigenvalue is derogatory if it is not nonderogatory. T We remark that an eigenvalue is nonderogatory if it is geometrically simple k and dim Ker(A l=1 l l I) (see [1, Cor. ] and [1, Thm. ]). Note that if the eigenvalue is nonderogatory then there is exactly one A-invariant subspace of dimension j for each j = 0; 1; : : : ; dim R (A). This follows from the denition of a nonderogatory eigenvalue and the fact that each invariant subspace is contained in some of the intersections in (1). If a simple rectiable contour l splits the spectrum of A l for l = 1; : : : ; k, then the Riezs projectors are dened by P (A l ; l ) := 1 i Z l (I A l ) 1 d;

3 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES l = 1; : : : ; k. They commute and we dene P (A; ) := P (A 1 ; 1 ) P (A k ; k ): The gap between the subspaces L and M in C n is dened by (L; M) = kp L P M k; where P L and P M are the orthogonal projectors on L and M, respectively. If L; M = f0g then (L; M) = max 8 < : sup xm kxk=1 d(x; L); sup xl kxk=1 d(x; M) (see Theorem in [, p. 88]). We say that an A{invariant subspace N is stable if for every > 0 there exists > 0 such that if B = fb 1 ; : : : ; B k g is a set of commuting matrices with ka l B l k < for l = 1; : : : ; k, then there exists a B{invariant subspace M such that (N ; M) : For comparison with our results we state Theorem of [, p. 8] that characterizes stable invariant subspaces for a single matrix. Theorem.1 (Gohberg, Lancaster, and Rodman) Suppose that 1 ; : : : ; r are all distinct eigenvalues of an n n matrix A over C. A subspace N of C n is A{invariant and stable if and only if N = N 1 _+ _+N r, where for each j the subspace N j is an arbitrary A{invariant subspace of R j (A) if dim Ker(I A) = 1, and either N j = f0g or N j = R j (A) if dim Ker(I A). Stability and root subspaces In this section we show that it suces to study the stability of invariant subspaces of root subspaces of A. The main result is that an A{invariant subspace N of C n is stable if and only if N is a direct sum N 1 _+ _+N r, where each N j is a stable A{invariant subspace of a root subspace of A. The following two lemmas are generalizations of Lemmas 15.. and 15.. of [, pp. 5-5]. The proofs are almost identical and therefore omitted. Lemma.1 Let i C i = 1; : : : ; k. Let be a simple rectiable contour that splits the spectrum of A i for P (A; ) = P (A 1 ; 1 ) P (A k ; k ) be the Riesz projector for A and = f 1 ; : : : ; k g and let A 0 restriction of A to Im P (A; ). Let N be a subspace of Im P (A; ). Then N is a stable invariant subspace for A if and only if N is a stable invariant subspace for A 0. 9 = ; () = fa 10 ; : : : ; A k0 g be the Lemma. Let N C n be an invariant subspace of A = fa 1 ; : : : ; A k g and assume that the contour i C splits the spectrum of A i for i = 1; : : : ; k. If N is stable for A then P (A; )N is a stable invariant subspace for the restriction A 0 = fa 10 ; : : : ; A k0 g of A to Im P (A; ).

4 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES Lemmas.1 and. imply the following theorem. Theorem. Let 1 = ( 11 ; : : : ; 1k ); : : : ; r = ( r1 ; : : : ; rk ) be all the dierent eigenvalues of a set of commuting matrices A = fa 1 ; : : : ; A k g. A subspace N of C n is A{ invariant and stable if and only if N = N 1 _+ _+N r, where N j is a stable A j {invariant subspace of the restriction A j = fa j1 ; : : : ; A jk g of A to R j (A) for j = 1; : : : ; r. Proof. Suppose that N is a stable A{invariant subspace. It is easy to see that N = N 1 _+ _+N r, where N j = N \ R j (A) for j = 1; : : : ; r. It follows from Lemma. that N j is a stable invariant subspace of the restriction A j for j = 1; : : : ; r. Next assume that each N j is a stable A j {invariant subspace. Lemma.1 implies that N j is a stable invariant subspace for A and therefore the direct sum N = N 1 _+ _+N r is a stable invariant subspace for A. Theorem. is similar to but weaker than Theorem.1 as it does not characterize the stable invariant subspaces. In particular, it is not yet clear which invariant subspaces of a root subspace at a derogatory eigenvalue are stable. Nevertheless, it enables us to study only the restriction of a set of commuting matrices to a root subspace. Now we are able to show that as it is the case for a single matrix a root subspace is a stable invariant subspace for a set of commuting matrices. Theorem. If = ( 1 ; : : : ; k ) is an eigenvalue of a set of commuting matrices A = fa 1 ; : : : ; A k g then the root subspace R (A) is a stable invariant subspace. Proof. Let i C be such closed contour that i lies inside i and all the other eigenvalues of A i lie outside i for i = 1; : : : ; k. It follows that the root subspace R (A) is equal to the image of the Riesz projector P (A; ) = P (A 1 ; 1 ) P (A k ; k ): Let B = fb 1 ; : : : ; B k g be a set of commuting matrices. If kb i A i k is suciently small then the matrix I B i is invertible for every i and the Riesz projector P (B i ; i ) is well dened. For each > 0 there exists > 0 such that if kb i A i k < then it follows kp (B i ; i ) P (A i ; i )k < (see [, p. 8] for details). The subspace Im P (B; ), where P (B; ) = P (B 1 ; 1 ) P (B k ; k ), is invariant for B. It is easy to see that for each > 0 there exists > 0 such that if kp (B i ; i ) P (A i ; i )k < for i = 1; : : : ; k then it follows that (Im P (B; ); Im P (A; )) < : As a consequence Im P (A; ) is a stable invariant subspace. Theorem. implies that it is enough to treat only sets of nilpotent commuting matrices. First we show that invariant subspaces of root subspaces of nonderogatory eigenvalues are stable. This also coincides with the theory for the single matrix case. A chain of subspaces f0g = M 0 M 1 M n = C n is called complete if dim M i = i for i = 0; 1; : : : ; n. It is well known fact that a set of commuting matrices is simultaneously similar to a set of upper-triangular commuting

5 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES 5 matrices. It follows then that for every set of commuting matrices there exists a complete chain of invariant subspaces. The following theorem is a generalization of Theorem 15.. of [, p. 9]. The proof is very similar and it is omitted. Theorem.5 Let A = fa 1 ; : : : ; A k g be a set of commuting matrices. For a given > 0, there exists > 0 such that the following holds: if B = fb 1 ; : : : ; B k g is such a set of commuting matrices that ka i B i k < for i = 1; : : : ; k and fm j g is a complete chain of B{invariant subspaces, then there exists a complete chain fn j g of A{invariant subspaces such that (N j ; M j ) < for j = 1; : : : ; n 1. Corollary. If 0 = (0; : : : ; 0) is a nonderogatory eigenvalue of a set of nilpotent commuting matrices A = fa 1 ; : : : ; A k g then each A{invariant subspace is stable. Proof. Since the eigenvalue 0 is nonderogatory the set A has only one j-dimensional invariant subspace N j for j = 0; 1; : : : ; n. (See the denition of a nonderogatory eigenvalue and the remark following it.) Subspaces N 0 ; : : : ; N n form a complete chain and we can apply Theorem.5. Corollary. Let A = fa 1 ; : : : ; A k g be a set of nilpotent commuting matrices. If N is the only A{invariant subspace of the dimension dim N, then N is a stable A{invariant subspace. Proof. Recall that there always exists a complete chain of invariant subspaces for A. Suppose that A has only one invariant subspace N of the dimension dim N. It follows then that the subspace N is a part of all complete chains of invariant subspaces. The result now follows from Theorem.5. A simple consequence of Corollary. is stability of the eigensubspace of a geometrically simple eigenvalue. The eigenvalue need not be nonderogatory and this result diers from the single matrix case. Namely, in the single matrix case, it follows that if an eigenspace is one-dimensional then the eigenvalue is nonderogatory and the stability follows by Theorem.1. On the other hand, in the case of a set of commuting matrices there exist eigenvalues that are geometrically simple and derogatory (see Example.9). Corollary.8 If = ( 1 ; : : : ; k ) is a geometrically simple eigenvalue of a set of commuting matrices A = fa 1 ; : : : ; A k g then the eigenspace Ker(A I) is a stable invariant subspace. Example.9 Suppose that n = and that e i, i = 1; ; ; are the standard basis vectors for C. Then A = ; B =

6 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES is a pair of commuting matrices that is geometrically simple and derogatory. By Corollary.8 the eigenspace L(e 1 ) is a stable invariant subspace. Here L(X) is the linear span of the set of vectors X. Assume that is a small positive number. Consider now two commuting perturbations: ; and ; It is easy to observe that L(e 1 ; e ) is the only two-dimensional invariant subspace for the rst perturbation and that L(e 1 ; e ) is the only two-dimensional invariant subspace for the second perturbation. Therefore the pair fa; Bg has no stable invariant subspace of dimension. If we take the transposed matrices A T = ; B T = then L(e +e ), for (; ) C nf0; 0g, are all the one-dimensional invariant subspaces of fa T ; B T g, while L(e ; e ) is the only two-dimensional invariant subspace. It follows by Corollary. that L(e ; e ) is stable. The above analysis of pair fa; Bg also shows that there is no stable one-dimensional invariant subspace for fa T ; B T g. The example L(e ; e ), which is a two-dimensional eigenspace for fa T ; B T g, shows that eigenspaces of dimension more than one can be stable invariant subspaces for sets of two or more commuting matrices. This diers from a single matrix case where it follows from Theorem.1 that all the eigenspaces of dimension two or more that are proper subspaces of a root subspace are unstable invariant subspaces. 5 5 : Problem.10 The main problem that remains open is to characterize all stable invariant subspaces of a k-tuple of nilpotent commuting matrices. Question.11 It is known that for a xed dimension d the variety of d-dimensional invariant subspaces of a single nilpotent matrix is connected [8, 1]. Is the variety of d-dimensional invariant subspaces of a k-tuple of nilpotent commuting matrices still connected? A pair of commuting matrices If the set contains only two commuting matrices, then we are able to show some additional results. First we show that although a pair of commuting matrices A and B may have innitely many invariant subspaces, it has only nitely many stable invariant subspaces. We use the fact that the set of pairs of commuting matrices where one of the matrices is nonderogatory is dense in the set of all pairs of commuting matrices. It was pointed out to us by one of the referees that this was an old result proved rst by Motzkin and

7 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES Taussky [15] and rediscovered several times. (See [].) We reproduce here a proof given by Guralnick []. We do so for the convinience of the reader and to facilitate the discussion on commuting triples of matrices. We say an n n matrix is generic if it has n distinct eigenvalues. Theorem.1 If fa; Bg is a pair of commuting nn matrices over C many stable invariant subspaces. then it has nitely Proof. It follows from Theorem. that it is enough to consider only a commuting pair of nilpotent matrices. If (0; 0) is a nonderogatory eigenvalue for fa; Bg, then there are only nitely many invariant subspaces which are all stable as a result of Corollary.. Thus we assume that (0; 0) is a derogatory eigenvalue. Let A = XJX 1, where J = diag(j n 1; : : : ; J nr ) is the Jordan canonical form for A. Since (0; 0) is a derogatory eigenvalue for fa; Bg, 0 is a derogatory eigenvalue for A and r. For distinct 1 ; : : : ; r the matrix R = X diag( 1 I n 1 + J n 1; : : : ; r I nr + J nr )X 1 is nonderogatory and commutes with matrix A. The matrix B = B + R () commutes with A for arbitrary C. Matrix B is nonderogatory except for nitely many values of. Therefore it is possible to choose arbitrary small > 0 such that B is nonderogatory. Assume now that B is nonderogatory. Then there exists a polynomial p such that A = p(b ). For an arbitrary > 0 we can approximate B with a generic matrix G such that kb Gk. Since A = p(b ), there exists > 0 such that ka p(g)k for kb Gk, i.e. pair (p(g); G) is close to pair (A; B). Since G is a generic matrix, it has only nitely many invariant subspaces and it follows that the pair fa; Bg has only nitely many stable invariant subspaces. Observe that in the above proof the polynomial p can be chosen so that both G and p(g) are generic. The following lemma shows that for a pair of commuting matrices stable invariant subspaces are determined by invariant subspaces of nearby generic commuting pairs. Lemma. Let fa; Bg be a pair of commuting nilpotent matrices and let N be an fa; Bg{invariant subspace. Then N is stable if and only if for every > 0 there exists > 0 such that if f e A; e Bg is a pair of generic commuting matrices with k e A Ak; k e B Bk < then there exists a f e A; e Bg{invariant subspace M such that (N ; M) : Proof. We only need to show that the condition is sucient for the stability of N. Suppose that for every > 0 there exists > 0 such that if f e A; e Bg is a pair of generic commuting matrices with k e A Ak; k e B Bk < then there exists a f e A; e Bg{invariant subspace M such that (N ; M) :

8 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES 8 Let N be an unstable invariant subspace for fa; Bg. Then for each m = 1; ; : : : there exist commuting pair fa m ; B m g and m > 0 such that ka m Ak; kb m Bk < 1=m and that (N ; M m ) + m for all invariant subspaces M m of fa m ; B m g. It follows from Theorem.5 that there exists # m > 0 such that # m < 1=m and that if f e Am ; e Bm g is a commuting pair that satises k e Am A m k; k e Bm B m k < # m then for each f e Am ; e Bm g{ invariant subspace e P there exist fam ; B m g{invariant subspace P such that (P; e P) m =: Since it is possible to nd a generic commuting pair arbitrarily close to the original commuting pair (see the proof of Theorem.1), for each m = 1; ; : : : this implies the existence of a generic commuting pair f e Am ; e Bm g such that k e Am Ak; k e Bm Bk < =m and that (N ; M m ) > for all invariant subspaces M m of f e Am ; e Bm g. This contradicts the initial assumption and thus it follows that N has to be a stable invariant subspace. Question. Is the set of stable invariant subspaces of any k-tuple (k ) of commuting matrices nite? Question. For a single matrix an invariant subspace is stable if and only if it corresponds to an isolated point of the variety of invariant subspaces. Is this the case also for a pair (or more generally for a k-tuple, k ) of commuting matrices? (See also Example..) Remark.5 If a set contains three or more commuting matrices then, in general, it is not possible to construct a nearby generic commutative set as it is done for pairs in the proof of Theorem.1. Suppose that we have a set of commuting matrices fa; B; Cg. If we follow the proof of Theorem.1 then it fails in the moment when we want to use the matrix B. This matrix commutes with A but not necessarily with C. Guralnick [] has even shown that in the general case of three or more commuting matrices it is not possible to approximate the set with a set of generic commuting matrices (see also [, 9, 10]). For this reason it is not possible, in general, to extend the proof of Theorem.1 to the commutative sets with more than two matrices. It follows from the results of Guralnick [] that the approximation for commuting k-tuples, k ; is possible if the size n of matrices is at most and is not possible in general if n. For triples of commuting matrices, it follows by results of Holbrook and Omladic in [10] that the approximation is possible if the size n is at most 5 and is not possible if n 0. For the remaining n, it is not known if the approximation is possible. The bounds for n in [10] are an improvement of bounds given earlier by Germanic [] and Guralnick and Sethuraman []. We conclude that the same arguments as in the proof of Theorem.1 show that if k = and n 5 or k and n then a k-tuple of commuting n n matrices has only nitely many stable invariant subspaces. Note that Lemma. can not be generalized to the arbitrary sets of three or more commuting matrices for the same reasons as Theorem.1.

9 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES 9 Example. Suppose that n = and that e i, i = 1; ; ; ; are the standard basis vectors for C. Let A = and B = be a pair of commuting matrices. It is geometrically simple and derogatory. By Corollary.8 the eigenspace L(e 1 ) is a stable invariant subspace. Recall that L(X) is the linear span of the set of vectors X. It is easy to show that two-dimensional invariant subspaces form the family L(e 1 ; e + e ) and three-dimensional invariant subspaces form the family L (e 1 ; e + e ; e + (e e )) ; where (; ) C nf(0; 0)g. Assume that is a small positive number. Consider now two commuting perturbations of fa; Bg: and ; 5 ; The rst perturbed pair is nilpotent and nonderogatory. Its complete chain of invariant subspaces is f0g L(e 1 ) L(e 1 ; e + e ) L(e 1 ; e + e ; e ) C : The second perturbed 5 5 : pair has four distinct eigenvalues. Corresponding joint eigenvectors are: ; 1 i i + i 5 5 ; 1 i i i 5 5 ; All its two-dimensional invariant subspaces are near the subspace L(e 1 ; e + e ) and all its three-dimensional subspaces are near the subspace L(e 1 ; e ; e ). These perturbations show that there is no three-dimensional stable invariant subspace for the pair fa; Bg. It also follows that two-dimensional invariant subspaces other than L(e 1 ; e + e ) are not stable. However, neither were we able to nd a commuting perturbation that would show that L(e 1 ; e + e ) is not stable, nor were we able to show that it is a stable invariant subspace. 5 : Remark. We observe that the subspace L(e 1 ; e + e ) in the above example is the only joint marked two-dimensional invariant subspace [, p. 8] for matrices A and B. Before we discuss this statement we give the denition of a marked invariant subspace. Let A be a n n matrix over C. The sequence of vectors x 1 ; : : : ; x k, x 1 = 0, such that (A xi 1 ; i = ; : : : ; k I)x i = 0 ; i = 1;

10 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES 10 is a Jordan chain of matrix A for the eigenvalue. Let N C n be an invariant subspace of A C nn. We say that N is marked if there exists a basis B = fx 11 ; x 1 ; : : : ; x 1n 1; x 1 ; : : : ; x n ; : : : ; x r1 ; : : : ; x rnr g () consisting of Jordan chains of A and there exist indexes 0 t j n j such that fx 11 ; x 1 ; : : : ; x 1t 1; x 1 ; : : : ; x t ; : : : ; x r1 ; : : : ; x rtr g is a basis for N. In other words, N is a marked invariant subspace of A if it is possible to choose its basis in such a way that it is extendable to a basis for C n consisting of Jordan chains of A. The notion of marked invariant subspace was rst dened by Gohberg, Lancaster, and Rodman [, p. 8]. See [] for an interesting characterization of marked invariant subspaces. Now we return to Example.. Observe that the subspaces L(e 1 ; e ) and L(e 1 ; e + e ) are the only two-dimensional marked invariant subspaces of A and that L(e 1 ; e ) and L(e 1 ; e + e ) are the only two-dimensional marked invariant subspaces of B. (See also [, Example.9.1, pp. 8-8].) More generally, suppose that fa; Bg is a pair of commuting nilpotent matrices. Suppose further that B is a Jordan basis for A given in () and that n j are chosen so that n 1 n n r 1. The Jordan basis B for A can be further chosen in such a way that vector Bx ij is in the span of vectors x kl with either l = j and k > i or l < n i j and k arbitrary (see e.g. the proof of Lemma in []). Let B be the matrix () dened in the same way as in the proof of Theorem.1. Recall from the proof there that for > 0 but small enough the matrix B is nonderogatory. It follows from our particular choice of the Jordan basis B that the spectrum of B is equal to f 1 ; ; : : : ; r g and that the multiplicity of the eigenvalue j is equal to n j. Recall from the proof of Theorem.1 that A = p(b ) for some polynomial p. Then it follows that each invariant subspace of B, and therefore also of the commuting pair fa; B g, is a marked invariant subspace of A. Thus, better understanding of the set of joint marked invariant subspaces of A and B might shed some light on problem of characterization of the set of stable invariant subspaces for pair fa; Bg. The above observation led as to pose the following question. Question.8 Is a stable invariant subspace of a pair of commuting matrices marked invariant subspace for each of the matrices? 5 Connection to algebraic multiparameter spectral theory In this section we study the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem. We consider an algebraic multiparameter system W: W i () = kx j=1 V ij j V i0 ; i = 1; ; : : : ; k; (k );

11 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES 11 where = ( 1 ; : : : ; k ) are parameters and V ij are n i n i matrices over C. The tensor product space C n 1 C n C n k is isomorphic to C N, where N = n 1 n n k. Linear transformations 0; 1; : : : ; k; and dened by and linearity. On C N and ij on C N are induced by V ij, i = 1; ; : : : ; k; j = ij (x 1 x x k ) = x 1 V ij x i x k we also dene operator determinants 0 = k 1 k. k1 11 1;i 1 1 ;i 1. k kk. 10 1;i+1 1k 0 ;i+1 k i =.... k1 k;i 1 k0 k;i+1 kk for i = 1; : : : ; k. A multiparameter system W is called nonsingular if the corresponding operator determinant 0 is invertible. In the case of a nonsingular multiparameter system W, we associate with W a k-tuple of commuting linear transformations = f 1 ; : : : ; k g, where i = 1 0 i, i = 1; : : : ; k (see [1, Thm...1]). An k-tuple C k is called an eigenvalue of the multiparameter system W if all W i () are singular. If Ker( I) := k\ i=1 Ker( i i I) = f0g; then is an eigenvalue of. Let (W) and ( ) denote the set of all the eigenvalues of W and, respectively. It was shown by Atkinson [1, Thm..9.1] that (W) = ( ) and that Ker( I) = Ker W 1 () Ker W () Ker W k (): An eigenvalue of a multiparameter system W is called nonderogatory [1] if is a nonderogatory eigenvalue of the associated system. We say that M C N is an invariant subspace for W if i M M; i = 1; : : : ; k: We say that an invariant subspace N of the multiparameter system (5) is stable if for a given > 0 there exists > 0 such that the following holds: if a nonsingular multiparameter system W 0 : W 0 i () = kx j=1 V 0 ij j V 0 i0 ; i = 1; ; : : : ; k; (5)

12 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES 1 is such that kv ij V 0 ijk < for all (i; j) then there exists an invariant subspace M of W 0 such that (N ; M) < : The stability is very important for the numerical calculation, for example, for the calculation of a basis for the root subspace of a nonderogatory eigenvalue [1, 1]. If the invariant subspace is not stable then we can not expect stable numerical calculation. Since i for i = 1; : : : ; k commute the stability of invariant subspaces for the algebraic multiparameter problem is closely related to the stability of invariant subspaces for commuting matrices. Multiparameter system W 0 is equivalent to the associated system 0 i x = ix; x = 0; i = 1; : : : ; k: () It is obvious that for each > 0 there exists > 0 such that if for all (i; j) then kv ij V 0 ijk < k i 0 ik < ; i = 1; : : : ; k: As a result we can apply a part of the theory on the stability of invariant subspaces of commuting matrices to the stability of invariant subspaces of multiparameter systems. The problems of stability are connected but not identical since in the study of stability for multiparameter eigenvalue problems we have to restrict the set of commuting matrices only to the matrices that form associated systems of multiparameter systems. For instance, let N be an invariant subspace of a multiparameter system W. If N is a stable invariant subspace for the commuting set = f 1 ; : : : ; k g, then N is also a stable invariant subspace of W. The converse is not necessarily true since an arbitrary set of commuting matrices is not necessarily an associated system of a multiparameter system. If we take for example matrices 1 = ; = ; 0 then 1 and are not associated with any multiparameter system (see [11, Example.1]). Summary of results that can be applied to the multiparameter eigenvalue problems is as follows. It follows from Theorem. that the complete root subspace is a stable invariant subspace. Corollary. yields that all invariant subspaces of root subspace of a nonderogatory eigenvalue are stable. This means that it is possible to numerically stable compute the basis for the root subspace of a nonderogatory eigenvalue [1, 1]. It also follows from Corollary.8 that the eigenspace of a geometrically simple eigenvalue is stable.

13 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES 1 New answers on the stability of invariant subspaces of multiparameter systems are connected with a study of conditions a set of commuting matrices = f 1 ; : : : ; k g must satisfy in order that there exists a multiparameter system W such that is its associated system. Some of the conditions are given in the preprint [1]. Acknowledgement. We wish to thank both referees for helpful suggestions. Specially, we wish to thank one of them for pointing out an error in the discussion of marked invariant subspaces in the version of the paper that we submitted rst. References [1] F. V. Atkinson, Multiparameter Eigenvalue Problems, Academic Press, New York, 19. [] V. Baranovsky, The Variety of Pairs of Commuting Nilpotent Matrices is Irreducible, Transformation Groups,, 001, -8. [] J. Ferrer, F. Puerta, and X. Puerta, Geometric Characterization and Classication of Marked Subspaces, Lin. Alg. Appl. 5, 199, 15-. [] I. Gohberg, P. Lancaster, and L. Rodman, Invariant Subspaces of Matrices with Application, Wiley{Interscience, New York, 198. [5] G. H. Golub and C. F. Van Loan, Matrix Computations, rd edition, The Johns Hopkins University Press, Baltimore, 199. [] R. M. Guralnick, A Note on Commuting Pairs of Matrices, Lin. Multilin. Alg. 1, 199, 1-5. [] R. M. Guralnick and B. A. Sethuraman, Commuting Pairs and Triples of Matrices and Related Varieties, Lin. Alg. Appl. 10, 000, [8] W. H. Gustafson and C. Martin, On Varieties of Invariant Subspaces, in Frequency Domain and State Space Methods for Linear Systems, C.I. Byrnes, A. Lindquist, eds., Elsevier Science Publ., 198, [9] J. Holbrook, Perturbation of Commuting Matrices, lecture given at LAW'9, Bled, Slovenia, 199. [10] J. Holbrook and M. Omladic, Approximating Commuting Operators, Lin. Alg. Appl., to appear. [11] T. Kosir, Commuting Matrices and Multiparameter Eigenvalue Problems, Ph.D. thesis, University of Calgary, Calgary, 199. [1] T. Kosir, On the Structure of Commutative Matrices, Lin. Alg. Appl. 18, 199, 1-18.

14 STABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES 1 [1] T. Kosir, Finite Dimensional Multiparameter Spectral Theory: The Nonderogatory Case, Lin. Alg. Appl. 1/1, 199, 5-0. [1] T. Kosir, Notes on Inverse Problems in Multiparameter Spectral Theory, preprint. [15] T. Motzkin and O. Taussky, Pairs of Matrices with Property L. II, Trans. Amer. Math. Soc. 80, 1955, [1] B. Plestenjak, A Numerical Algorithm for Computing a Basis for the Root Subspace at a Nonderogatory Eigenvalue of a Multiparameter System, Lin. Alg. Appl. 85, 1998, 5{. [1] M. A. Shayman, On the Variety of Invariant Subspaces of a Finite-Dimensional Linear Operator, Trans. Amer. Math. Soc., 198, 1-.