Matrix best approximation in the spectral norm

TL;DR

This paper develops a matrix formulation for spectral approximation problems, relates them to semidefinite programming, and provides geometric characterizations and numerical methods, with applications to iterative linear algebra methods.

Contribution

It introduces a new matrix formulation of spectral approximation, links it to semidefinite programming, and offers geometric insights and numerical solutions, extending previous convergence analyses.

Findings

Derived a matrix formulation of spectral approximation

Connected spectral approximation to semidefinite programming

Established conditions for equality of min-max and max-min values

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.