ABSOLUTE VARIATION OF RITZ VALUES, PRINCIPAL ANGLES AND SPECTRAL SPREAD

P. MASSEY , D. STOJANOFF Y S, ZÁRATE

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS

SIAM PUBLICATIONS

Lugar: Philadelphia-USA; Año: 2021 vol. 42 p. 1506 - 1527

0895-4798

Let $A$ be a $dimes d$ complex self-adjoint matrix, $mathcal{X},mathcal{Y}subset mathbb{C}^d$ be $k$-dimensional subspaces and let $X$ be a $dimes k$ complex matrix whose columns form an orthonormal basis of $mathcal{X}$; that is, $X$ is an isometry whose range is the subspace $cX$.We construct a $dimes k$ 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 Knyazev and Argentati.