or H = UU = nx i=1 i u i u i ; where H is a non-singular Hermitian matrix of order n, = diag( i ) is a diagonal matrix whose diagonal elements are the

Transcription

1 Relative Perturbation Bound for Invariant Subspaces of Graded Indenite Hermitian Matrices Ninoslav Truhar 1 University Josip Juraj Strossmayer, Faculty of Civil Engineering, Drinska 16 a, Osijek, Croatia, truhar@gfos.hr Ivan Slapnicar 1 University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, R. Boskovica b.b, 1000 Split, Croatia, ivan.slapnicar@fesb.hr Abstract We give a bound for the perturbations of invariant subspaces of graded indenite Hermitian matrix H = D AD which is perturbed into H+H = D (A+A)D. Such relative perturbations include important case where H is given with an element-wise relative error. Application of our bounds reuires only the knowledge of the size of relative perturbation kak, and not the perturbation A itself. This typically occurs when data are given with relative uncertainties, when the matrix is being stored into computer memory, and when analyzing some numerical algorithms. Subspace perturbations are measured in terms of perturbations of angles between subspaces, and our bound is therefore relative variant of the well-known Davis{Kahan sin theorem. Our bounds generalize some of the recent relative perturbation results. 1 Introduction and preliminaries We are considering the Hermitian eigenvalue problem Hu i = i u i ; 1 Authors acknowledge the grant from the Croatian Ministry of Science and Technology. Part of this work was done while the second author was visiting the Pennsylvania State University, Department of Computer Science and Engineering, University Park, PA, USA. Preprint submitted to Elsevier Science 14 April 1999

2 or H = UU = nx i=1 i u i u i ; where H is a non-singular Hermitian matrix of order n, = diag( i ) is a diagonal matrix whose diagonal elements are the eigenvalues of H, and U = [ u 1 u u n ] is unitary matrix whose i-th column is an eigenvector which corresponds to i. Subspace X is an invariant subspace of a general matrix H if HX X. If H is Hermitian and the set of eigenvalues f i1 ; i ; ; ik g does not intersect the rest of the spectrum of H, then the corresponding k- dimensional invariant subspace is spanned by the eigenvectors u i1 ; u i ; ; u ik. Throughout the paper kk and kk F will denote the -norm and the Frobenius norm, respectively. Our aim is to give bound for perturbations of invariant subspaces for the case when H is a graded Hermitian matrix, that is, H = D AD; (1) where D is some non-singular grading matrix, under Hermitian relative perturbation of the form H + H f H = D (A + A)D: Our bound is a relative variant of the well-known sin theorems by Davis and Kahan [], [4, Section V.3.3]. The development of relative perturbation results for eigenvalue and singular value problems has been very active area of research in the past years [3,1,4,9,1,6,5,15,16,7,13] (see also the review article [1]). We shall rst describe the relative perturbation and state the existing eigenvalue perturbation results. Let H be the Hermitian relative perturbation which satises jx Hxj x H x; 8x; < 1; () where H = p H = UjjU is a spectral absolute value of H (that is, H is the positive denite polar factor of H). Under such perturbations the relative change in eigenvalues is bounded by [9] 1? e j j 1 + : (3)

3 This ineuality implies that the perturbations which satisfy () are inertia preserving. This result is very general and includes important classes of perturbations. If H is a graded matrix dened by (1) and ba = D? H D?1 ; (4) then (3) holds with = kakk b A?1 k: (5) Indeed, jx Hxj = jx D ADxj = kx D ADxk kx D kkakkdxk kakk b A?1 k x H x; (6) as desired. Another important class of perturbations is when H is perturbed element-wise in the relative sense, jh ij j "jh ij j: (7) By setting D = diag( H ii ); (8) since ja ij j "ja ij j, the relation (6) implies that (3) holds with = "kjajkk b A?1 k: (9) Since b A ii = 1, we have k b A?1 k ( b A) nk b A?1 k, where (A) kakka?1 k is the spectral condition number. Also, kjajk n (see [9, Proof of Theorem.16]). The diagonal grading matrix D from (8) is almost optimal in the sense that [3] ( b A) n min D ( D H D) n( H ) = n(h); where the minimum is taken over all non-singular diagonal matrices. Similarly, for more general perturbations of the type jh ij j "D ii D jj ; (10) In [9] no attention was paid to scalings with non-diagonal matrix D, so this result is new. Also notice that any perturbation H + H can clearly be interpreted as the perturbation of a graded matrix, and vice versa. 3

4 (3) holds with = "nk b A?1 k "n( b A): (11) Remark 1 Application of the bounds (5), (9) and (11) reuires only the knowledge of the size of relative perturbation kak, and not the perturbation A itself. Such situation occurs in several cases which are very important in applications. When the data are determined to some relative accuracy or when the matrix is being stored in computer memory, the only available information about H is that it satises (7). Similarly, in error analysis of various numerical algorithms (matrix factorizations, eigenvalue or singular value computations), the the only available information is that H satises more general condition (10). If H is positive denite, then H = H, and (9) and (11) reduce to the corresponding results from [4]. We would like to point out a major dierence between positive denite and indenite case. Remark If H is positive denite, then small perturbations of the type (7) and (10) cause small relative changes in eigenvalues if and only if ( b A) (A) is small [4]. If H is indenite, then from (9) and (11) it follows that small ( b A) implies small relative changes in eigenvalues. However, these changes can be small even if ( b A) is large [9,18]. Although such examples can be successfully analyzed by perturbations through factors as in [9,8], this shows that the graded indenite case is essentially more dicult than the positive denite one. Perturbation bounds for eigenvectors of simple eigenvalues were given for scaled diagonally dominant matrices in [1], and for positive denite matrices in [4]. The bound for invariant subspace which corresponds to single, possibly multiple, eigenvalue of an indenite Hermitian matrix was given in [9, Theorem.48]. This bound, given in terms of projections which are dened by Dunford integral as in [14, Section II.1.4], is generalized to invariant subspaces which correspond to a set of neighboring eigenvalues in [5]. In [16, Theorems 3.3 and 3.4] two bounds for invariant subspaces of a graded positive denite matrix were given. We generalize the result of [16, Theorem 3.3] and give the bound for k sin k F for a graded non-singular indenite matrix. Our results are, therefore, relative variants of the well-known sin theorems [, Section ], [4, Theorem V.3.4]. Our results also generalize the ones from [9,5] to subspaces which correspond to arbitrary set of eigenvalues. Our results and the related results from [1,4,9,1] and other works, are also useful in estimating the accuracy of highly accurate algorithms for computing eigenvalue decompositions [1,4,7,18]. The paper is organized as follows: in Section we prove our main theorem. In Section 3 we show how to eciently apply that theorem, in particular for the 4

5 important types of relative perturbations (7) and (10) (c.f. Remarks 1 and ). We also describe some classes of well-behaved indenite matrices. In Section 4 we give some concluding remarks. Bound for invariant subspaces Let the eigenvalue decomposition of the graded matrix H be given by H D AD = UU = [ U 1 U ] ; (1) 0 U U where 1 = diag( i1 ; i ; ; ik ); = diag( j1 ; j ; ; jl ); and k + l = n. We assume that 1 and have no common eigenvalues, thus, U 1 and U both span simple invariant subspaces according to [4, Denition V.1.]. Similarly, let the eigenvalue decomposition of H f be fh D (A + A)D = U e e U e = [ U e 1 U e e 1 0 U e 1 ] 0 e eu ; (13) where e 1 = diag( e i1 ; ; e ik ) and e = diag( e j1 ; ; e jl ). Let XY be a singular value decomposition of the matrix e U U 1. The diagonal entries of the matrix sin, are the sines of canonical angles between subspaces which are spanned by the columns of U 1 and e U 1 [4, Corollary I.5.4]. Before stating the theorem, we need some additional denitions. Let A = QQ = Qjj 1= Jjj 1= Q ; (14) be an eigenvalue decomposition of A. Here J is diagonal matrix of signs whose diagonal elements dene the inertia of A and, by Sylvester's theorem [10, Theorem 4.5.8], of H, as well. Set G = D Qjj 1= ; (15) such that H = GJG. Further, set N = jj?1= Q AQjj?1= ; (16) 5

6 such that f H = G(J + N)G. Finally, the hyperbolic eigenvector matrix of a matrix pair (M; J), where M is a Hermitian positive denite matrix, and J = diag(1), is the matrix X which simultaneously diagonalizes the pair such that X MX = M and X JX = J, where M is a positive denite diagonal matrix. Some properties of hyperbolic eigenvector matrices will be discussed in the next section. Theorem 3 Let H and f H be given as above, and let ka?1 kkak < 1. Then H and f H have the same inertia, thus e U 1 and e U from (13) span simple invariant subspaces, as well, and the canonical angles between the subspaces spanned by U 1 and e U 1 are bounded by k sin k F k e U U 1 k F ka?1 k kak 1 F? ka?1 k kak kv 1 k kv e k? e ; (17) j j min 1pk 1l e j j provided that the above minimum is greater than zero. Here V = [ V 1 V ] is the hyperbolic eigenvector matrix of the pair (G G; J), where G is dened by (14) and (15), and e V = [ e V 1 e V ] is the hyperbolic eigenvector matrix of the pair ([(I + NJ) 1= ] G G(I + NJ) 1= ; J); where N is dened by (16). V and e V are partitioned accordingly to (1) and (13). Note that the matrix suare root exists since by the assumption knk ka?1 k kak < 1 (see [11, Theorem 6.4.1]). Proof. The proof is similar to the proof of [16, Theorem 3.3]. From knk < 1 we conclude that the matrices H = GJG, J + N and f H = G(J + N)G all have the same inertia dened by J. Therefore, (1) and (13) can be written as H = UJU ; f H = e U e J e U ; (18) with the same J = diag(1) as in (14), respectively. Also, J + N can be decomposed as J + N = (I + NJ) 1= J[(I + NJ) 1= ] ; (19) which follows from (I + NJ) 1= = J[(I + NJ) 1= ] J. Thus, we can write f H as fh = G(I + NJ) 1= J[(I + NJ) 1= ] G : (0) 6

7 From (18) and H = GJG we conclude that the matrix G has the hyperbolic singular value decomposition [17] given by G = Ujj 1= JV J; V JV = J: (1) Similarly, from (18) and (0) we conclude that the matrix G(I + JN) 1= has the hyperbolic singular value decomposition given by G(I + NJ) 1= = e U j e j 1= J e V J; e V J e V = J: () Now (0) and (19) imply that fh? H = G(I + NJ) 1= J[(I + NJ) 1= ] G? GJG = G(I + NJ) 1= G ; (3) where = J[(I + NJ) 1= ]? (I + NJ)?1= J = (I + NJ)?1= N: (4) Pre- and post-multiplication of (3) by e U and U, respectively, together with (1), (13), (1) and (), gives e e U U? e U U = j e j 1= J e V J JV Jjj 1= : This, in turn, implies e e U U 1? e U U 1 1 = j e j 1= J e V J JV 1 J 1 j 1 j 1= ; where J = J 1 J is partitioned accordingly to (1) and (13). By interpreting this euality component-wise we have [ U e U 1 ] p = [J V e j e ; jj 1;pp j J JV 1 J 1 ] p ; e ;? 1;pp for all p f1; : : : ; kg and f1; : : : ; lg. By taking the Frobenius norm we have k e U U 1 k F kj e V J JV 1 J 1 k F max p; j e ; jj 1;pp j e ;? 1;pp : 7

8 Further, kj e V J JV 1 J 1 k F k e V k kv 1 k kk F : The relations (4), (16) and the assumption imply kk F k(i + JN)?1= k knk F 1 1? knk knk F ka?1 k kak F 1? ka?1 k kak; and the theorem follows by combining the last three relations. For positive denite H the matrices V e and V are unitary. Thus kv 1 k = kv e k = 1, and Theorem 3 reduces to [16, Theorem 3.3]. Thus, the dierence from the positive denite and indenite case is existence of the additional factor kv 1 k kv e k in the indenite case. This again shows that the indenite case is essentially more dicult (c.f. Remark ). The minimum in (17) plays the role of the relative gap, although the function j? e j= j e j does not necessarily increase with the distance between and e if they have dierent signs. For example, if 1 = f1g and e = f?1; 0:1g, then the minimum is attained between 1 and?1 and not between 1 and 0:1. Some other results for positive denite matrices and singular value decomposition from [15,16] can be generalized to indenite case and hyperbolic singular value decomposition, respectively, by using the techniues similar to those used in the proof of Theorem 3 and Section 3. As already mentioned in Section 1, Theorem 3 generalizes some other relative bounds from [1,4,9,1], as well. The bound (17) involves unperturbed and the perturbed uantities ( and e, V 1 and e V ), and can, therefore be computed only if A is known. The existing bounds for graded indenite Hermitian matrices from [9,1,5], depend neither upon perturbed vectors nor eigenvalues, and are therfore simpler to compute, and can be applied to some important problems where only the size of the relative perturbation is known (c.f. Remark 1). Our next aim is to remove the dependence on the perturbed uantities from (17), and to explain when is the factor kv 1 k k e V k in (17) expected to be small. We do this in the next section. 8

9 3 Applying the bound We show how to remove the perturbed uantities from the bound of Theorem 3. In particular, we show how to eciently compute and the upper bound for the factor kv 1 k k e V k. In Section 3.1 we show that this factor will be small if, for the chosen grading D, the matrix b A which is dened by (4) is well conditioned. We also describe some easily recognisable classes of matrices which fulll this condition for any diagonal grading D. First note that the perturbed eigenvalues can be expressed in terms of the unperturbed one by using (3) and (5), that is, the minimum in (17) is bounded by min 1pk 1l? e j j e j j min 1pk 1l? j (1 + sign( i? j ) sign( j ))j : (5) j (1 + )j We now proceed as follows: we rst bound k e V k in terms of kv k; we then bound kv k in terms of ka?1 k and k b Ak; and, nally, we show how to eciently compute k b Ak and from (5), (9) or (11). The matrices for which X JX = J, where J = diag(1) are called J-unitary. Such matrices have the following properties: { XJX = J. { kxk = kx?1 k. Moreover, the singular values of X come in pairs of reciprocals, f; 1=g. { Let J = I l (?I n?l ) (6) and let X be partitioned accordingly, X11 X X = 1 : X X 1 Then kx 11 k = kx k, kx 1 k = kx 1 k, and kxk = kx 1 k kx 1 k = kx 1 k + kx 11 k: (7) These eualities follow from the CS decomposition of X [0]. Lemma 4 Let J be given by (6) and let X and f X be two J-unitary matrices which are partitioned accordingly in block columns as X = [ X p X n ] and f X = [ f Xp f Xn ] : 9

10 X p spans the so called positive subspace with respect to J since XpJX p = I l. Similarly, X n spans the negative subspace. Then the matrix X JX f is also J-unitary, and kx J f Xk = kx nj f X p k kx nj f X p k: Also, k f Xk kxnj X f p k kxnj X f p k kxk: (8) Proof. The euality follows from (7), and the ineuality follows since k f Xk = kx? X J f Xk kx? k kx J f Xk = kxk kx J f Xk. Lemma 5 Let X and f X be the hyperbolic eigenvector matrices of the pairs (M; J) and ( f M; J) where M and f M are positive denite, f M = (I +?) M(I +?), and J is given by (6). Let X and f X be partitioned as in Lemma 4. Dene = k?k F =(1? k?k). If kxk < 1 4 ; (9) then k f Xk kxk : 1? 4kXk Proof. The fact that X diagonalizes the pair (M; J) can be written as X MX = p n ; (30) where is diagonal positive denite matrix which is partitioned according to J and X. J-unitarity of X and (30) imply MX = X? = JXJ, thus MX p = JX p p and MX n =?JX n n : (31) Relations analogous to (30) and (31) hold for f M, f X and e, as well. By premultiplying f M f X p = J f X p e p by X n we have, after using (31) and rearranging, n X nj f X p + X nj f X p e p = X n( f M? M) f X p : 10

11 Set b? = I? (I +?)? (the inverse exists since (9) implies k?k < 1). By using the identity f M? M = b? f M + M?, the above euality can be written component-wise as [X nj f X p ] ij ([ n ] ii + [ e p ] jj ) = [X n ] :i fb? f M + M?g[ f X p ] :j = [X n ] :i b?j[ f X p ] :j [ e p ] jj? [ n ] ii [X n ] :ij?[ f X p ] :j : Here [X] :k denotes the k-th column of the matrix X. By dividing this euality by [ n ] ii + [ e p ] jj and by using the fact that maxf[ n ] ii ; [ e p ] jj g [ n ] ii + [ e p ] jj < 1; we have j[x nj f X p ] ij j j[x n ] :i b?j[ f X p ] :j j + j[x n ] :ij?[ f X p ] :j j: Since k b?k F, by taking Frobenius norm we have kx nj f X p k F kx n k k f X p k(k b?k F + k?k F ) kxk k f Xk: By inserting this ineuality in (8), we have k f Xk kxk k f Xk (kxk k f Xk) kxk: After a simple manipulation we obtain k f Xk kxk 1? 4kXk : as desired. Now we can use Lemma 5 to bound kv e k by kv k in Theorem 3. V diagonalizes the pair (G G; J) where G is dened by (15), and V e diagonalizes the pair ([(I + NJ) 1= ] G G(I + NJ) 1= ; J), where N is dened by (16). In order to apply Lemma 5 with M = G G and I +? = (I +NJ) 1= we need to bound k?k in terms of knk. Since knjk < 1, we can apply Taylor series of the function (1 + x) 1= to the matrix (I + NJ) 1= [11, Theorem 6..8], which gives 1X (I + NJ) 1= = I + n=1 n?1 (n? 3)!! (?1) (NJ) n n I +?: n! 11

12 Here (n? 3)!! = (n? 3). By taking norms we have k?k 1X n=1 1 knk 1X (n? 3)!! n knk n = 1 1X n! knk n=1 n=1 knk n?1 1 knk 1? knk : Similarly, for the Frobenius norm we have k?k F 1 knk F 1? knk : (n? 3)!! knk n?1 n?1 n! From the above two ineualities we see that from Lemma 5 is bounded by knk F? 3kNk ka?1 k kak F : (3)? 3kA?1 k kak If ka?1 k kak F < 4kV k + 3 ; (33) then < 1=(4kV k ), that is, (9) holds and Lemma 5 gives k e V k kv k : (34) 1? 4kV k We can further bound kv k in terms of A and b A from (4). By denition, V diagonalizes the pair (G G; J) where H = GJG, that is, V G GV = jj, where is the eigenvalue matrix of H. Then, the eigenvalue decomposition of H is given by H = UU where U = GV jj?1=. Then H = UjjU = GV V G : (35) Therefore kv k = kv V k = kg?1 H G? k. In our case, from (4) and (15), we have kv k = kjj?1= Q D? D b ADD?1 Qjj?1= k ka?1 k k b Ak: (36) We can now prove the following theorem. 1

13 Theorem 6 Let the assumptions of Theorem 3 hold and let, in addition, ka?1 k kak F < =(4kA?1 k k b Ak + 3). Then k sin k F ka?1 k kak 1 F? ka?1 k kak ka?1 k kak 1 F? ka?1 k kak min 1pk 1l min 1pk 1l 1? e j j e j j 1? e j j e j j kv k 1? 4kV k ka?1 k k b Ak 1? 4kA?1 k k b Ak (37) (38) where is dened by (3). Proof. The assumption and (36) imply (33), which in turn implies (34). Now (37) follows by inserting (34) in Theorem 3, and (38) follows by inserting (36) in (37). Note that the assumption of the theorem implies the positivity of the second suare root in (38), and is therefore not too restrictive provided that ka?1 k and k b Ak are not too large. Some classes of matrices which fulll this condition are described in Section 3.1. Also, instead of (36) we can use an alternative bound kv k = kv?1 k kak k b A?1 k: (39) In order to apply (38), besides ka?1 k, kak F and kak, we also need to know k b Ak. For the special case (8) when D is diagonal, we simply have k b Ak trace( b A) = n. Such D appears naturally when we consider perturbations of the type (7) and (10) which occur in numerical computation (see Remark 1). We now show that, for any D, k b Ak from (38) and from (5) can be computed by highly accurate eigenreduction algorithm from [7,18] at little extra cost 3. This algorithm rst factorizes H as H = F JF by symmetric indenite factorization [19]. This factorization is followed by one-sided J-orthogonal Jacobi method on the pair F; J. This method forms the seuence of matrices F k+1 = F k X k ; where X kjx k = J: 3 In [18] only the algorithm for the real symmetric case was analyzed, but a version for the Hermtian case is possible, as well. 13

14 This seuence converges to some matrix F X which has numerically orthogonal columns, and F is J orthogonal, that is, F JF = J. The eigenvalues of H are approximated by = J diag(x F F X), and the eigenvectors are approximated by U = F Xjj?1=. Therefore, H = F XX F. Note that X also diagonalizes the pair (F F; J). Since the matrix F X is readily available in the computer, we can compute b A as ba = D? F XX F D?1 ; (40) or, even simpler, just its factor D? F X. Finally, after computing b A, can be computed directly from the denitions (5), (9), or (11), respectively. 3.1 "Well-behaved" matrices Our bounds dier from the bounds for the positive denite case [16, Theorem 3.3] by additional factors, namely the last uotients in (37) and (38). From (36) and (39) we also have kv k ((A)( b A)) 1=. From (1) and (4) and the denitions of H and H from Section 1, we have A = D? Ujj 1= Jjj 1= U D?1 ; b A = D? Ujj 1= jj 1= U D?1 : This implies that (see also [9, Proof of Theorem.16]) ja ij j A b iiajj b ; ja?1 ij j A b?1 ii ba?1 jj ; and hence k jaj k trace( b A); k ja?1 j k trace( b A?1 ): This implies that A is well conditioned if so is b A. We conclude that the additional factors will be small if, for the chosen grading D, any of the right hand sides in (36) or (39) are small, or (which is a more restrictive condition) simply if b A is well conditioned. We call such matrices \well-behaved". We also conclude that an invariant subspace is stable under small relative perturbations for the chosen grading D, if ka?1 k is small, any of the three above conditions is fullled, and the eigenvalues which dene the subspace are well relatively separated from the rest of the spectrum. Although b A can be easily computed as described in the comments of Theorem 6, we are interested in identifying classes of \well-behaved" matrices in advance. In the positive denite case the answer is simple { such are the well 14

15 graded matrices, that is the matrices of the form H = DAD, where D is diagonal positive denite such that A ii = 1, and A is well conditioned. In the indefinite case we have two easily recognisable classes of \well-behaved" matrices, namely scaled diagonally dominant matrices [1] and Hermitian uasidenite matrices [6,8]. Moreover, for the same A such matrices are well-behaved for any diagonal grading D. Scaled diagonally dominant matrices have the form H = D(J + )D, where D is diagonal positive denite, J = diag(1) and k k < 1. For these matrices A = J + and kak k b A?1 k n(1 + k k)=(1? k k) [9, Theorem.9]. Thus, (39) implies that the last uotient in (37) is small when k k is not too close to one. Also, kak which is used in the denition of in (3) has to be suciently small. Hermitian matrix H is uasidenite if there exists a permutation P such that P T HP = H = H 11 H 1 H ; 1? H where H 11 and H are positive denite. The proof that such matrix is \wellbehaved" is rather involved. Quasidenite matrix always has a triangular factorization H = F JF, where F is lower triangular with real diagonal and J is diagonal with J ii = sign(h ii ) [6] 4. Set H = DAD where D = diag(jh ii j). Then H = D A D = D A 11 A 1 A 1? A D; D = P T DP; A = P T AP: (41) Note that A is also uasidenite, and its triangular factorization is given by A = BJB where B = D?1 F. Let us now bound ( b A). Assume without loss of generality that the diagonal elements of D are nonincreasing which can be easily attained by permutation. From (40) we have ( b A) (B) (X) ; (4) where, as already mentioned, X diagonalizes the pair (F F; J). In [] it was shown that (X) min ( F F ); 4 The proof in [6] is for the real symmetric case, but it is easily seen that it holds for the Hermitian case, as well. 15

16 where the minimum is over all matrices which commute with J. In our case this clearly implies kxk (F D?1 ) = (DBD?1 ): Since B is lower triangular and D has nondecreasing diagonal elements, we have jdbd?1 j jbj and jdb?1 D?1 j jb?1 j, so that (X) k jbj k k jb?1 j k kbk F kb?1 k F F (B): (43) The matrix JA is positive denite according to [9]. By appropriately modifying the proof of [9, Theorem], we have k jbj jbj T k F n(kt k + kst?1 Sk); where T = [JA + (JA) T ]= and S = [JA? (JA) T ]=. Note that the proof in [9] is for the real case, but it readily holds in the complex case as is recently shown in [8]. From (41) we have k jbj jbj T k F n(kp T T P k + kp T SP P T T?1 P P T SP k) (44) n maxfk A 11 k + k A 1 A?1 A k; 1 k A k + k A 1A?1 11 A 1 k: Now note that the inverse of a uasidenite matrix is also uasidenite [6, Theorem 1.1]. By modifying the proof of [9, Theorem] we obtain a bound similar to (44) for k jb?t j jb?1 jk F (for details see [8]). By combining this discussion with (43), from (4) we conclude that essentially ( A) b will be small if in (41) A 11 and A are well conditioned and k A 1 k is not too large. 4 Conclusion We derived new perturbation bounds for invariant subspaces of non-singular Hermitian matrices. Our bounds improve the existing bounds in the following aspects. Our bounds extend the bounds for scaled diagonally dominant matrices from [1] to general indenite matrices. Our bounds also extend the bound of [16, Theorem 3.3] for positive denite matrices to indenite case. Finally, our bounds extend the bounds for indenite matrices from [9,5] to subspaces which correspond to any set of possibly nonadjacent eigenvalues (numerical experiments also indicate that our bounds tend to be sharper). For graded matrices of the form (1) which are well-behaved according to Section 3.1, our 16

17 bounds are sharper that the general bound for diagonalizable matrices from [13], k sin k F kh?1 Hk F min 1pk 1l?e j j j : (45) This bound makes no preference for positive denite matrices over indenite ones, and is easier to interpret than our bound. However, since in (45) H is multiplied by H?1 from the left, this bound does not accommodate two sided grading well. Finally, our bounds are computable from unperturbed uantities and can be eciently used in analyzing numerical algorithms in which graded perturbations naturally occur. We would like to thank the referee for valuable remarks which lead to considerable improvements in the paper. References [1] J. Barlow and J. Demmel, Computing accurate eigensystems of scaled diagonally dominant matrices, SIAM J. Numer. Anal., 7:76{791 (1990). [] C. Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal., 7:1{46 (1970). [3] J. Demmel and W. M. Kahan, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Stat. Comput., 11:873{91 (1990). [4] J. Demmel and K. Veselic, Jacobi's method is more accurate than QR, SIAM J. Matrix Anal. Appl., 13:104{144 (199). [5] Z. Drmac, Computing the Singular and the Generalized Singular Values, PhD thesis, Fernuniversitat, Hagen, [6] S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation techniues for singular value problems, SIAM J. Numer. Anal., 3:197{1988 (1995). [7] S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, BIT, 38:50{509 (1998). [8] P. E. Gill, M. A. Saunders and J. R. Shinnerl, On the stability of Cholesky factorization for symmetric uasidenite systems, SIAM J. Matrix Anal. Appl, 17:35{46 (1996). [9] G. H. Golub and C. F. Van Loan, Unsymmetric positive denite linear systems, Linear Algebra Appl., 8:85{97 (1979). 17

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