Yimin Wei a,b,,1, Xiezhang Li c,2, Fanbin Bu d, Fuzhen Zhang e. Abstract

Transcription

1 Linear Algebra and its Applications 49 (006) wwwelseviercom/locate/laa Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices Application of perturbation theory for simple invariant subspaces Yimin Wei ab Xiezhang Li c Fanbin Bu d Fuzhen Zhang e a School of Mathematical Sciences Fudan University Shanghai PR China b Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University) Ministry of Education Shanghai PR China c Department of Mathematical Sciences Georgia Southern University Statesboro GA USA d Institute of Mathematics Fudan University Shanghai PR China e Division of Math Science and Technology Nova Southeastern University 330 College Avenue Fort Lauderdale FL 3334 USA Received 7 July 005; accepted 6 June 006 Available online 4 August 006 Submitted by S Kirkland Abstract Perturbation bounds for the relative error in the eigenvalues of diagonalizable and singular matrices are derived by using perturbation theory for simple invariant subspaces of a matrix and the group inverse of a matrix These upper bounds are supplements to the related perturbation bounds for the eigenvalues of diagonalizable and nonsingular matrices 006 Elsevier Inc All rights reserved AMS classification: 5A60; 65F35 Keywords: Diagonalizable and singular matrix; Simple invariant subspaces; Separation function; Relative perturbation bound Corresponding author Address: School of Mathematical Sciences Fudan University Shanghai PR China addresses: ymwei@fudaneducn (Y Wei) xli@georgiasouthernedu (X Li) fanbinbu@yahoocomcn (F Bu) zhang@novaedu (F Zhang) The work of this author was supported by the National Natural Science Foundation of China under grant and Shanghai Education Committee Partial work of this author was finished when the author visited the Key Laboratory of Mathematics for Nonlinear Sciences of Fudan University /$ - see front matter ( 006 Elsevier Inc All rights reserved doi:006/jlaa

2 766 Y Wei et al / Linear Algebra and its Applications 49 (006) Introduction Let A be an n n diagonalizable complex matrix with an eigendecomposition A = X A ΛXA where Λ = diag{λ λ λ n } and X A is an eigenmatrix of A Assume that E is a perturbation matrix of A and that µ is an eigenvalue of A + E The well-known Bauer Fike theorem [] reveals a bound for the absolute error between µ and the closest eigenvalue of A by min λ i µ κ(x A ) E i where κ(x A ) = X A XA is the condition number of X and is the matrix -norm Under an additional assumption that A is nonsingular Eisenstat and Ipsen [4] showed the following relative perturbation bound λ i µ min κ(x A ) A E () i λ i They concluded that relative perturbation bounds are not necessarily stronger than absolute bounds and the conditioning of an eigenvalue in the relative sense is the same as in the absolute sense Recently the conclusion has been extended to the case of singular and diagonalizable matrices by Eisenstat [3] In that paper the author developed an upper bound for µ>κ(x A ) E (ie µ is too large in magnitude for the zero eigenvalue of A to satisfy the Bauer Fike bound) λ i µ min + α λ i /=0 λ i κ(x A ) A # E () / where α = κ(x A ) E µ (κ(x A ) E ) and A # denotes the group inverse of A From () the author also derived a uniform upper bound min λ i /=0 λ i µ λ i κ(x A ) A # E (3) for µ κ(x A ) E In this paper a completely new approach through perturbation theory of simple invariant subspaces is developed to derive a relative perturbation bound which is an improvement on our earlier results [50] Some preliminary results about the perturbation theory of simple invariant spaces of a matrix and the separation function of two square matrices are introduced in Section A relative perturbation bound with respect to norms for nonzero eigenvalues of diagonalizable and singular matrices is derived with an application of perturbation theory of simple invariant spaces in Section 3 Preliminaries Let A C n n be a diagonalizable and singular matrix with rank(a) = r<n There are from [6 Theorem 55] is a unitary matrix [X Y ] with [X X ] =[Y Y ] such that the spectral resolution of A is given by Y C O A[X X ]= () O O n r Y where C C r r is nonsingular and O n r is the zero matrix of order n r The group inverse A # (see [8]) is given by [ A # C O Y ] =[X X ] () O O n r Y

3 Y Wei et al / Linear Algebra and its Applications 49 (006) Let E be a perturbation matrix of A AsE approaches zero each eigenvalue of A + E approaches the corresponding eigenvalue of A Then a relative perturbation bound is meaningful only for those nonzero eigenvalues of A On the other hand we would estimate an upper bound of those eigenvalues of A + E which approach zero For this purpose we will apply perturbation theory of simple invariant subspaces of a square matrix due to Stewart and Sun [6 Chapter V] Writing [ Y Y we have [Y Y ] E[X X ]= ] (A + E)[X X ]= F F (3) F F C + F F F F It follows that X Y = AA # X Y = I AA # C = Y A# X F ij = Yi EX j ij = Recall [X Y ] is a unitary matrix With the notations above the following relations are easily derived from () to(4): AA # = I AA # if A is a nonzero singular matrix (see [9]) C = Y A # X = X Y A# X = A # X A # F AA # E F = AA # E(I AA # ) (5) F E F E(I AA # ) We next introduce simple invariant subspaces of A and the separation function (see [6 Chapter V] and [7]) It follows from () that AX = X C and AX = X O n r Let σ(m) denote the spectrum of a square matrix M Since σ(c) and σ(o n r ) are disjoint the ranges of X and X are called simple invariant subspaces of A The separation function of two arbitrary square matrices S and is defined as { min W = WS W sep (S ) := if σ(s) σ() = 0 otherwise With notations in () and (3) it follows from [6 Theorem 58] that if sep (C O n r ) F F > F F there is a unique matrix P satisfying P < such that [ Y Y PY F sep (C O n r ) F F (4) (6) ] C + F + F (A + E)[X + X P X ]= P F (7) O F PF

4 768 Y Wei et al / Linear Algebra and its Applications 49 (006) Relative bounds with respect to norms In this section we derive relative perturbation bounds for the eigenvalues of diagonalizable and singular matrix with an application of perturbation theory of simple invariant subspaces Lemma 3 Let C be nonsingular and let O be a square null matrix Then sep (C O) = C (3) Proof The result follows directly from the definition of sep (C O) and the fact that WC C W with equality for a W whose rows are multiples of left singular vectors corresponding to the smallest singular value of C Eq (3) will be used in the proof of the following main result of this paper Theorem 3 Let A be a diagonalizable and singular matrix and let E be a perturbation matrix If E < 4 A # (3) AA # then for µ σ(a+ E) either µ E(I AA # ) + p A # E(I AA # ) (33) or λ µ min κ (X A ) { A # E 0 /=λ σ (A) λ + p A # E(I AA # ) } (34) where X A is an eigenmatrix of A and E p = A # AA # E E(I AA # ) Proof Suppose that A C n n has a eigendecomposition XA D O AX A = O O n r where D is nonsingular diagonal matrix of order r r n and X A is an eigenmatrix of A Let X A =[S S ] and XA =[T T ] where S T C n r Suppose that S = X R and T = Y R are QR decompositions of S and T respectively We denote X := S R and Y := T R It follows that [X X ] =[Y Y ] [X Y ] =[X Y ] and the spectral resolution of A is given by Y A[X X ]= Y C O (35) O O where C = R DR is nonsingular So X C :=R is an eigenmatrix of C Moreover we have κ(x C ) κ(x A ) (36)

5 Y Wei et al / Linear Algebra and its Applications 49 (006) In fact X A =[X X C X R ] and XA =[Y XC Y R ] which implies that X C = X X C X A XC = XC Y X XC Y X = Y XC XA With the assumption (3) and relations in (5) it is obvious that E < 4 A # AA # 4 C AA # It follows from relations in (5) and (3) that F + F + F F 4 E AA # < C = sep (C O) Then according to (7) there exists a unique matrix P satisfying F P < C F F E A # AA # E E(I AA # ) = p (37) such that C + F + F (A + E) similar to P F O F PF So σ(a+ E) = σ(f PF ) σ(c + F + F P) We first consider the case when µ σ(f PF ) It follows from (5) and (37) that µ F PF F + P F E(I AA # ) + p AA # E(I AA # ) (38) which completes the proof of (33) For the second case when µ σ(c + F + F P) According to (35) the set of all nonzero eigenvalues of A and the spectrum of C are identical With Eisenstat and Ipsen s result in () and (36) abovewehave λ µ λ µ min = min κ(x C ) C (F + F P) 0 /=λ σ (A) λ λ σ(c) λ κ(x A ) C (F + F P) (39) Furthermore it follows from (4) and (5) that ) C (F + F P) = Y A# X (Y EX + Y EX P A # E + P A # E(I AA # ) Thus (34) follows by replacing P with its upper bound in (37) We complete the proof of the theorem

6 770 Y Wei et al / Linear Algebra and its Applications 49 (006) Remark There is a tighter bound than (34) for small P Writing C = CX C XC C C and using [4 Theorem 3] and the fact that C C = XC CX C = D is diagonal we have λ µ min C λ σ(c) λ (F + F P)C = XC C (F + F P)X C But with the notation above X C C (F + F P)X C = R ) Y A# X (Y EX + Y EX P R ( ) = R Y A# AA # E X R I + X R R PR = T A # E [S S ] I R PR Thus λ µ λ µ min = min X A 0 /=λ σ (A) λ λ σ(c) A # λ E X A + R PR κ(x A ) A # E ( + ) κ(x A) P (30) since R PR T P S κ(x A ) P We remark that for a sufficiently small E the upper bounds in (34) and (30) are dominated by κ(x A ) A # E The term κ(x A ) could be interpreted as an approximate condition number of the relative perturbation error for eigenvalues of A up to the first order of E Inequality () gives an upper bound for a specific µ which is known (3) gives a simple uniform upper bound owever it follows from (37) that if A # AA # E < 4 then p AA # < which implies that A # E + p A # E(I AA # ) < A # E Thus in this case (34) gives a better bound than (3) In an analogous way if P is small ( ) κ(x such that κ(x A ) P < then + A ) P < Thus in this case (30) gives a better bound than (3) Acknowledgements The authors thank Professor S Kirkland and referees for their valuable suggestions and very detailed comments on earlier drafts of the manuscript The first author would like to thank Professors Ren-cang Li and Qiang Ye for their useful discussions and hospitality during his visit at University of Kentucky in 000 References [] F Bauer C Fike Norms and exclusion theorems Numer Math (960) 37 4 [] A Ben-Israel TNE Greville Generalized Inverses: Theory and Applications second ed Springer-Verlag New York 003

7 Y Wei et al / Linear Algebra and its Applications 49 (006) [3] SC Eisenstat A perturbation bound for the eigenvalues of a singular diagonalizable matrix Linear Algebra Appl 46 (006) [4] SC Eisenstat ICF Ipsen Three absolute perturbation bounds for matrix eigenvalues imply relative bounds SIAM J Matrix Anal Appl 0 (998) [5] X Li Y Wei A note on the relative perturbation bounds of the eigenvalues of a diagonalizable matrix a talk presented at the Tenth International Linear Algebra Society (ILAS) Conference Auburn University Alabama USA June [6] GW Stewart JG Sun Matrix Perturbation Theory Academic Press Boston 990 [7] GW Stewart Error bounds for approximate invariant subspaces of closed linear operators SIAM J Numer Anal 8 (97) [8] G Wang Y Wei S Qiao Generalized Inverses: Theory and Computations Science Press Beijing 004 [9] Y Wei On the perturbation of the group inverse and oblique projection Appl Math Comput 98 (999) 9 4 [0] Y Wei F Zhang Problem 8-9: A relative perturbation bound IMAGE The Bull Int Linear Algebra Soc (ILAS) 8 (April) (00) 35; Solution to Problem 8-9 IMAGE The Bull Int Linear Algebra Soc (ILAS) 9 (October) (00) 33 [] Y Wei X Li F Bu F Zhang Relative perturbation bounds for the eigenvalues of singular matrices August 005 Preprint Available from: < buxxx00/publications/r_perturbationpdf>