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arXiv · 2506.09687

Matrix best approximation in the spectral norm

Vance Faber·Jörg Liesen·Petr Tichý
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Abstract

We derive, similar to Lau and Riha, a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the relation of the spectral approximation problem to semidefinite programming, and we present a simple MATLAB code to solve the problem numerically. We then obtain geometric characterizations of spectral approximations that are based on the $k$-dimensional field of $k$ matrices, which we illustrate with several numerical examples. The general spectral approximation problem is a min-max problem, whose value is bounded from below by the corresponding max-min problem. Using our geometric characterizations of spectral approximations, we derive several necessary and sufficient as well as sufficient conditions for equality of the max-min and min-max values. Finally, we prove that the max-min and min-max values are always equal when we ``double'' the problem. Several results in this paper generalize results that have been obtained in the convergence analysis of the GMRES method for solving linear algebraic systems.

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Vance Faber, Jörg Liesen, Petr Tichý. 2025-06-11. Matrix best approximation in the spectral norm. https://arxiv.org/abs/2506.09687

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