Abstract
The paper contains two parts. First, by applying the results about the eigenvalue perturbation bounds for Hermitian block tridiagonal matrices in paper [1], we obtain a new efficient method to estimate the perturbation bounds for singular values of block tridiagonal matrix. Second, we consider the perturbation bounds for eigenvalues of Hermitian matrix with block tridiagonal structure when its two adjacent blocks are perturbed simultaneously. In this case, when the eigenvalues of the perturbed matrix are well-separated from the spectrum of the diagonal blocks, our eigenvalues perturbation bounds are very sharp. The numerical examples illustrate the efficiency of our methods.
Highlights
There are many known results about eigenvalue perturbation bounds of Hermitian matrices
By comparing the differences in the equation (3.5) with the bounds obtained by the Theorem 3.1, we can find that the singular values perturbation bounds obtained by the Theorem 3.1 are sharp and this estimating method is efficient
On the basis of conclusions of the paper [1], we study eigenvalue perturbation bounds of block tridiagonal matrix for the cases where two adjacent blocks of A are perturbed and the first s blocks of A are perturbed by the perturbation matrix E1,s
Summary
There are many known results about eigenvalue perturbation bounds of Hermitian matrices. For Hermitian matrices with special sparse structures such as block tridiagonal Hermitian matrix, the Weyl’s theorem may not be the best choice. If we apply the results above repeatedly, we can obtain a weaker upper bounds Inspired by these questions, in this paper, we expect to obtain the perturbation bounds for singular value of a block tridiagonal matrix.
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