Convexity of sums of eigenvalues of a segment of unitaries

TL;DR

This paper proves the convexity of sums of eigenvalue angles for certain paths of unitary matrices near the identity, with applications to unitarily invariant and Finsler norms in matrix groups.

Contribution

It establishes the convexity of sums of eigenvalue angles along specific matrix paths and characterizes when these sums are linear, extending to applications in invariant norms.

Findings

Convexity of sum of eigenvalue angles near the identity matrix.

Linear sum of angles implies commutativity of matrices.

Extension of results to Finsler norms in special unitary groups.

Abstract

For a $n\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\le m\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path $t\mapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided the path stays in a vecinity of the identity matrix). This vecinity is described in terms of the opertor norm of matrices, and it is optimal. We show that the when all the maps $t\mapsto s_m(t)$ are linear, then $x$ commutes with $y$. Several application to unitarily invariant norms in the unitary group are given. Then we extend these applications to $Ad$-invariant Finsler norms in the special unitary group of matrices. This last result is obtained by proving that any $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the supremum of a family of what we call orbit norms, induced by the Killing form of $K$.