# 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 \((\Vert R_i^*R_j\Vert _{\mathop {\mathrm{tr}}\,})_{1 \le i, j \le 3}\) for any rectangular complex (or real) matrices \(R_1, R_2, R_3\) so that the multiplication \(R_i^*R_j\) is compatible for all \(i, j,\) where \(\Vert \cdot \Vert _{\mathop {\mathrm{tr}}\,}\) denotes the trace norm. We then give a complete analysis of the problem when the trace norm is replaced by other unitarily invariant norms.

## Keywords

Polar decomposition Positive semi-definite matrix Trace norm Unitarily invariant norm## Mathematics Subject Classification

Primary 15A60 15B47 Secondary 47A30 47B65## Notes

### Acknowledgments

Li was an affiliate member of the Institute for Quantum Computing, University of Waterloo, an honorary professor of the Shanghai University, and University of Hong Kong. The research of Li was supported by the USA NSF and HK RCG. This project was done while Li and Zhang were participating the 2014 Summer International Program at the Shanghai University. Li and Zhang would like to thank the support of the Shanghai University.

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