Positivity of Partitioned Hermitian Matrices with Unitarily Invariant Norms

Abstract

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

6 pages

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