A Note on Eigenvalues of Perturbed Hermitian Matrices
نویسندگان
چکیده
Let A = ( H1 E ∗ E H2 ) and à = ( H1 O O H2 ) be Hermitian matrices with eigenvalues λ1 ≥ · · · ≥ λk and λ̃1 ≥ · · · ≥ λ̃k, respectively. Denote by ‖E‖ the spectral norm of the matrix E, and η the spectral gap between the spectra of H1 and H2. It is shown that |λi − λ̃i| ≤ 2‖E‖ η + √ η2 + 4‖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.
منابع مشابه
Departure from Normality and Eigenvalue Perturbation Bounds
Perturbation bounds for eigenvalues of diagonalizable matrices are derived that do not depend on any quantities associated with the perturbed matrix; in particular the perturbed matrix can be defective. Furthermore, Gerschgorin-like inclusion regions in the Frobenius are derived, as well as bounds on the departure from normality. 1. Introduction. The results in this paper are based on two eigen...
متن کاملComputing the Matrix Geometric Mean of Two HPD Matrices: A Stable Iterative Method
A new iteration scheme for computing the sign of a matrix which has no pure imaginary eigenvalues is presented. Then, by applying a well-known identity in matrix functions theory, an algorithm for computing the geometric mean of two Hermitian positive definite matrices is constructed. Moreover, another efficient algorithm for this purpose is derived free from the computation of principal matrix...
متن کاملSome results on the polynomial numerical hulls of matrices
In this note we characterize polynomial numerical hulls of matrices $A in M_n$ such that$A^2$ is Hermitian. Also, we consider normal matrices $A in M_n$ whose $k^{th}$ power are semidefinite. For such matriceswe show that $V^k(A)=sigma(A)$.
متن کاملProof of a trace inequality in matrix algebra
for all complex vectors ai and bj . One can easily prove that if X is positive definite then X is hermitian (see, e.g., Ref. [1], p. 65). Since the eigenvalues of hermitian matrices are real, it is easy to prove that the eigenvalues of positive definite matrices are real and positive. Moreover, a positive definite matrix is invertible, since it does not possess a zero eigenvalue. Note that a no...
متن کاملA Note on Computing Eigenvalues of Banded Hermitian Toeplitz Matrices
It is pointed out that the author's O(n 2) algorithm for computing individual eigenvalues of an arbitrary n × n Hermitian Toeplitz matrix T n reduces to an O(rn) algorithm if T n is banded with bandwidth r.
متن کامل