A refined bound for the spectral variations of matrices

Gil’, Michael

doi:10.14232/actasm-020-566-z

Cited by 2 publications

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“…In the appropriate situations it considerably improves Elsner's inequality and the main result from [18]. In this section we present the main results from [19].…”

Section: A Bound For the Spectral Variation Via The Entries Of Matricesmentioning

confidence: 76%

“…In the paper [18], a bound for the spectral variation has been explicitly expressed via the entries of the considered matrices. In the paper [19], we have established a new bound via the entries. In the appropriate situations it considerably improves Elsner's inequality and the main result from [18].…”

Section: A Bound For the Spectral Variation Via The Entries Of Matricesmentioning

confidence: 99%

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Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey

Gil’

2021

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The paper is a survey of the recent results of the author on the perturbations of matrices. A part of the results presented in the paper is new. In particular, we suggest a bound for the difference of the determinants of two matrices which refines the well-known Bhatia inequality. We also derive new estimates for the spectral variation of a perturbed matrix with respect to a given one, as well as estimates for the Hausdorff and matching distances between the spectra of two matrices. These estimates are formulated in the terms of the entries of matrices and via so called departure from normality. In appropriate situations they improve the well-known results. We also suggest a bound for the angular sectors containing the spectra of matrices. In addition, we suggest a new bound for the similarity condition numbers of diagonalizable matrices. The paper also contains a generalization of the famous Kahan inequality on perturbations of Hermitian matrices by non-normal matrices. Finally, taking into account that any matrix having more than one eigenvalue is similar to a block-diagonal matrix, we obtain a bound for the condition numbers in the case of non-diagonalizable matrices, and discuss applications of that bound to matrix functions and spectrum perturbations. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the traditional methods and results.

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“…In the appropriate situations it considerably improves Elsner's inequality and the main result from [18]. In this section we present the main results from [19].…”

Section: A Bound For the Spectral Variation Via The Entries Of Matricesmentioning

confidence: 76%