The minimization of matrix logarithms - on a fundamental property of the unitary polar factor
Johannes Lankeit, Patrizio Neff, Yuji Nakatsukasa · arXiv (Cornell University) · 2013
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.