The minimization of matrix logarithms - on a fundamental property of the unitary polar factor

TL;DR

This paper proves that the unitary polar factor uniquely minimizes the matrix logarithm norm of a nonsingular matrix across all unitaries, using advanced inequalities and majorization theory.

Contribution

It establishes the fundamental minimization property of the unitary polar factor for matrix logarithms, a new theoretical insight in matrix analysis.

Findings

U is the unique minimizer for ||Log(Q*Z)|| over unitary Q.

U minimizes ||sym(Log(Q*Z))|| for any invertible Z.

The proof employs generalized Bernstein trace inequality and majorization theory.

Abstract

We show that the unitary factor U in the polar decomposition of a nonsingular matrix Z = U H is the minimizer for both ||Log(Q^* Z)|| and ||sym (Log(Q^*Z))|| over unitary Q, for any given invertible complex n-times-n matrix Z, for any unitarily invariant norm and any n. We prove that U is the unique matrix with this property. As important tools we use a generalized Bernstein trace inequality and the theory of majorization.