Positivity of Partitioned Hermitian Matrices with Unitarily Invariant Norms

TL;DR

This paper provides a concise proof of a recent positivity result for certain 3x3 matrices formed from trace norms of matrix products, and extends the analysis to other unitarily invariant norms.

Contribution

It offers a simplified proof of Drury's positivity result and thoroughly analyzes the positivity conditions under various unitarily invariant norms.

Findings

Confirmed positivity for matrices with trace norm

Extended positivity analysis to other unitarily invariant norms

Provided complete characterization of positivity conditions

Abstract

We give a short proof of a recent result of Drury on the positivity of a $3\times 3$ matrix of the form $(\|R_i^*R_j\|_{\rm tr})_{1 \le i, j \le 3}$ for any rectangular complex (or real) matrices $R_1, R_2, R_3$ so that the multiplication $R_i^*R_j$ is compatible for all $i, j$, where $\|\cdot\|_{\rm tr}$ denotes the trace norm. We then give a complete analysis of the problem when the trace norm is replaced by other unitarily invariant norms.