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Absolute variation of Ritz values, principal angles, and spectral spread. (English) Zbl 1479.15024

Summary: Let \(A\) be a \(d\times d\) complex self-adjoint matrix, let \(\mathcal{X},\mathcal{Y}\subset\mathbb{C}^d\) be \(k\)-dimensional subspaces, and let \(X\) be a \(d\times k\) complex matrix whose columns form an orthonormal basis of \(\mathcal{X}\); that is, \(\mathcal{X}\) is an isometry whose range is the subspace \(\mathcal{X}\). We construct a \(d\times k\) complex matrix \(Y_r\) whose columns form an orthonormal basis of \(\mathcal{Y}\) and obtain sharp upper bounds for the singular values \(s(X^*AX-Y_r^*\,A\,Y_r)\) in terms of submajorization relations involving the principal angles between \(\mathcal{X}\) and \(\mathcal{Y}\) and the spectral spread of \(A\). We apply these results to obtain sharp upper bounds for the absolute variation of the Ritz values of \(A\) associated with the subspaces \(\mathcal{X}\) and \(\mathcal{Y}\) that partially confirm conjectures by A. V. Knyazev and M. E. Argentati [SIAM J. Matrix Anal. Appl. 29, No. 1, 15–32 (2006; Zbl 1196.15016); Linear Algebra Appl. 415, No. 1, 82–95 (2006; Zbl 1092.65030)].

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A42 Inequalities involving eigenvalues and eigenvectors
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References:

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