A Note on Eigenvalues of Perturbed Hermitian Matrices

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1 A Note on Eigenvalues of Perturbed Hermitian Matrices Chi-Kwong Li Ren-Cang Li July 2004 Let ( H1 E A = E H 2 Abstract and à = ( H1 H 2 be Hermitian matrices with eigenvalues λ 1 λ k and λ 1 λ k, respectively. Denote by E the spectral norm of the matrix E, and η the spectral gap between the spectra of H 1 and H 2. It is shown that λ i λ i 2 E 2 η + η E 2, which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations. AMS Classifications: 15A42, 15A18, 65F15. Keywords: Hermitian matrix, eigenvalue, singular value. 1 Introduction Consider a partitioned Hermitian matrix A = ( m n m H 1 E n E H 2, (1.1 where E is E s complex conjugate transpose. At various situations (typically when E is small, one is interested in knowing the impact of removing E and E on the eigenvalues Department of Mathematics, The College of William and Mary, Williamsburg, Virginia Research supported in part by the NSF grant DMS Department of Mathematics, University of Kentucky, Lexington, KY This work was supported in part by the National Science Foundation CAREER award under Grant No. CCR

2 of A. More specifically, one would like to obtain bounds for the differences between that eigenvalues of A and those of its perturbed matrix à = ( m n m H 1 n H 2. (1.2 Let λ(x be the spectrum of the square matrix X, and let Y be the spectral norm of a matrix Y, i.e., the largest singular value of Y. There are two kinds of bounds for the eigenvalues λ 1 λ m+n and λ 1 λ m+n of A and Ã, respectively: 1. [1, 7, 8] λ i λ i E. ( [1, 2, 3, 5, 7, 8] If the spectra of H 1 and H 2 are disjoint, then where and λ(h i is the spectrum of H i. λ i λ i E 2 /η, (1.4 η def = min µ 1 λ(h 1, µ 2 λ(h 2 µ 1 µ 2, The bounds of the first kind do not use information of the spectral distribution of the H 1 and H 2, which will give (much weaker bounds when η is not so small; while the bounds of the second kind may blow up whenever H 1 and H 2 have a common eigenvalue. Thus both kinds have their own drawbacks, and it would be advantageous to have bounds that are always no bigger than E, of ( E as η 0, and at the same time behave like ( E 2 /η for not so small η. To further motivate our study, let us look at the following 2 2 example. Example 1 Consider the 2 2 Hermitian matrix ( α ɛ A =. (1.5 ɛ β Interesting cases are when ɛ is small, and thus α and β are approximate eigenvalues of A. We shall analyze by how much the eigenvalues of A differ from α and β. Without loss of generality, assume α > β. The eigenvalues of A, denoted by λ ±, satisfy λ 2 (α + βλ + αβ ɛ 2 = 0; and thus λ ± = α + β ± (α + β 2 4(αβ ɛ 2 2 = α + β ± (α β 2 + 4ɛ

3 Now { λ+ α 0 < β λ } = (α β + (α β 2 + 4ɛ 2 2 = 2ɛ 2 (α β + (α β 2 + 4ɛ 2 (1.6 which provides a difference that enjoys the following properties: 2ɛ 2 ɛ always, (α β + ɛ as α β +, (α β 2 + 4ɛ 2 ɛ 2 /(α β. The purpose of this note is to extend this 2 2 example and obtain bounds which improve both (1.3 and (1.4. Such results are not only of theoretical interest but also important in the computations of eigenvalues of Hermitian matrices [4, 6, 9]. As an application, similar bounds are presented for the singular value problem. 2 Main Result Theorem 2 Let A = ( m n m H 1 E n E H 2 and à = ( m n m H 1 n H 2 be Hermitian matrices with eigenvalues λ 1 λ 2 λ m+n and λ1 λ 2 λ m+n, (2.1 respectively. Define η i η def = min λ i µ 2, if λ i λ(h 1, µ 2 λ(h 2 min λ i µ 1, if λ i λ(h 2, µ 1 λ(h 1 (2.2 def = min η i = min µ 1 µ 2. (2.3 1 i m+n µ 1 λ(h 1, µ 2 λ(h 2 Then for i = 1, 2,, m + n, we have λ i λ i 2 E 2 (2.4 η i + η 2i + 4 E 2 2 E 2 η + η E 2. (2.5 3

4 Proof. Suppose U H 1 U and V H 2 V are in diagonal form with diagonal entries arranged in descending order. We may assume that U = I m and V = I n. therwise, replace A by (U V A(U V. We may perturb the diagonal of A so that all entries are distinct, and apply continuity argument for the general case. We prove the result by induction on m + n. If m + n = 2, the result is clear (from our Example. Assume that m + n > 2, and the result is true for Hermitian matrices of size m + n 1. First, refining an argument of Mathias [5], we show that (2.4 holds for i = 1. Assume that the (1, 1 entry of H 1 equals λ 1. By the min-max principle [1, 7, 8], we have λ 1 e 1Ae 1 = λ 1, where e 1 is the first column of the identity matrix. Let ( I X = m 0 (H 2 µ 1 I n 1. E I n Then where ( X H1 (λ (A λ 1 IX = H 2 λ 1 I n, H 1 (λ 1 = H 1 λ 1 I m E (H 2 λ 1 I n 1 E. Since A and X AX have the same inertia, we see that H 1 (λ 1 has zero as the largest eigenvalue. Notice that the largest eigenvalue of H 1 λ 1 I is λ 1 λ 1 0. Thus, for δ 1 = λ 1 λ 1 = λ λ 1, we have (see [7, (10.9] λ 1 λ 1 + E 2 2/(δ 1 + η 1, and hence Consequently, δ 1 E 2 /(δ 1 + η 1. 2 E δ 1 η 1 + η E 2 as asserted. Similarly, we can prove the result if the (1, 1 entry of H 2 equals λ 1. In this case, we will apply the inertia arguments to A and Y AY with ( I Y = m 0 E(H 1 λ 1 I m 1. I n Applying the result of the last paragraph to A, we see that (2.2 holds for i = m + n. 4

5 Now, suppose 1 < i < m + n. The result trivially holds if λ i = λ i. Suppose λ i λ i. We may assume that λ i > λ i. therwise, replace (A, Ã, i by ( A, Ã, m + n i + 1. Delete the row and column of A that contain the diagonal entry λ n. Suppose the resulting matrix  has eigenvalues ν 1 ν m+n 1. By the interlacing inequalities [7, Section 10.1], we have λ i ν i and hence λi λ i λ i νi. (2.6 Note that λ i is the ith largest diagonal entries in Â. Let η i be the minimum distance between λ i and the diagonal entries in the diagonal block Ĥj in  not containing λ i ; here j {1, 2}. Then η i η i because Ĥj may have one fewer diagonal entries than H j. Let Ê be the off-diagonal block of Â. Then Ê E. Thus, λ i λ i = λ i λ i because λ i > λ i λ i ν i by (2.6 2 Ê 2 by induction assumption η i + η 2i + 4 Ê 2 2 Ê 2 because η i η i η i + η 2i + 4 Ê 2 = 1 ηi Ê 2 η i 1 ηi E 2 η i because Ê E = 2 E 2 η i + ηi E 2 as asserted. 3 Application to Singular Value Problem In this section, we apply the result in Section 2 to study singular values of matrices. For notational convenience in connection to our discussion, we define the sequence of singular values of a complex p q matrix X by σ(x = (σ 1 (X,..., σ k (X, where k = max{p, q} and σ 1 (X σ k (X are the nonnegative square roots of the eigenvalues of the matrix XX or X X depending on which one has a larger size. Note that the nonzero eigenvalues of XX and X X are the same, and they give rise to the nonzero singular values of X which are of importance. We have the following result concerning the nonzero singular values of perturbed matrices. 5

6 Theorem 3 Let B = ( k l m G 1 E 1 n E 2 G 2 ( m G and B = 1 n G 2 k l be complex matrices with singular values σ 1 σ 2 σ max{m+n,k+l} and σ 1 σ 2 σ max{m+n,k+l}, (3.1 respectively, so that G 1 and G 2 are non-trivial. Define ɛ = max{ E 1, E 2 }, and min σ i µ 2, if σ i σ(g 1, µ 2 σ(g 2 η i η def = min σ i µ 1, if σ i σ(g 2, µ 1 σ(g 1 (3.2 def = min η i = min µ 1 µ 2. (3.3 1 i m+n µ 1 σ(g 1, µ 2 σ(g 2 Then for i = 1, 2,, min{m + n, k + l}, we have σ i σ i 2ɛ 2 (3.4 η i + η 2i + 4ɛ2 2ɛ 2 η + η 2 + 4ɛ 2, (3.5 and σ i = σ i = 0 for i > min{m + n, k + l}. Proof: By Jordan-Wielandt Theorem [8, Theorem I.4.2], the eigenvalues of ( B B are ±σ i and possibly some zeros adding up to m+n+k+l eigenvalues. A similar statement holds for B. Permuting the rows and columns appropriately, we see that ( G 1 E 1 B B is similar to G 1 E2 E 2 G 2, E1 G 2 and ( B B is similar to G 1 G 1 G 2 G 2. 6

7 Applying Theorem 2 with ( Gi H i = G i ( E2 and E =, E 1 we get the result. ne can also apply the above proof to the degenerate cases when G 1 or G 2 in the matrix B is trivial, i.e., one of the parameters m, n, k, l is zero. These cases are useful in applications. We state one of them, and one can easily extend it to other cases. Theorem 4 Suppose B = ( G E and B = ( G are p q matrices with singular values σ 1 σ max{p,q} and σ 1... σ max{p,q}, respectively. Then for i = 1,..., min{p, q}, σ i σ i 2 E. 2 σ i + σ 2i + 4 E 2 Acknowledgment This paper evolved from an early note of the second author in which Theorem 2 was proven for special cases. The authors wish to thank Editor Volker Mehrmann for making this collaboration possible. 7

8 References [1] R. Bhatia. Matrix Analysis. Graduate Texts in Mathematics, vol Springer, New York, [2] J. Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, [3] G. H. Golub and C. F. van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, Maryland, 2nd edition, [4] W. Kahan. When to neglect off-diagonal elements of symmetric tri-diagonal matrices. Technical Report CS42, Computer Science Department, Stanford University, [5] R. Mathias. Quadratic residual bounds for the hermitian eigenvalue problem. SIAM Journal on Matrix Analysis and Applications, 19: , [6] C. C. Paige. Eigenvalues of perturbed Hermitian matrices. Linear Algebra and its Applications, 8:1 10, [7] B. N. Parlett. The Symmetric Eigenvalue Problem. SIAM, Philadelphia, This SIAM edition is an unabridged, corrected reproduction of the work first published by Prentice-Hall, Inc., Englewood Cliffs, New Jersey, [8] G. W. Stewart and J.-G. Sun. Matrix Perturbation Theory. Academic Press, Boston, [9] J. H. Wilkinson. The Algebraic Eigenvalue Problem. Clarendon Press, xford, England,

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