A bivariate extension of the Crouzeix-Palencia result with an application to Fr\'echet derivatives of matrix functions

Michel Crouzeix; Daniel Kressner

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arXiv · 2007.09784

A bivariate extension of the Crouzeix-Palencia result with an application to Fr\'echet derivatives of matrix functions

Michel Crouzeix·Daniel Kressner

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Abstract

A result by Crouzeix and Palencia states that the spectral norm of a matrix function $f(A)$ is bounded by $K = 1+\sqrt{2}$ times the maximum of $f$ on $W(A)$, the numerical range of $A$. The purpose of this work is to point out that this result extends to a certain notion of bivariate matrix functions; the spectral norm of $f\{A,B\}$ is bounded by $K^2$ times the maximum of $f$ on $W(A)\times W(B)$. As a special case, it follows that the spectral norm of the Fr\'echet derivative of $f(A)$ is bounded by $K^2$ times the maximum of $f^\prime$ on $W(A)$. An application to the convergence analysis of certain Krylov subspace methods and the extension to functions in more than two variables are discussed.

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BibTeX RIS

Michel Crouzeix, Daniel Kressner. 2020-07-19. A bivariate extension of the Crouzeix-Palencia result with an application to Fr\'echet derivatives of matrix functions. https://arxiv.org/abs/2007.09784

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