The minimization of matrix logarithms: On a fundamental property of the unitary polar factor

We show that the unitary factor UpUp in the polar decomposition of a nonsingular matrix Z=UpHZ=UpH is a minimizer for both‖Log(Q⁎Z)‖and‖sym⁎(Log(Q⁎Z))‖ over the unitary matrices Q∈U(n)Q∈U(n) for any given invertible matrix Z∈Cn×nZ∈Cn×n, for any unitarily invariant norm and any n  . We prove that UpUp is the unique matrix with this property to minimize all these norms simultaneously. As important tools we use a generalized Bernstein trace inequality and the theory of majorization.