# Positivity of partitioned Hermitian matrices with unitarily invariant norms

Positivity of partitioned Hermitian matrices with unitarily invariant norms We give a short proof of a recent result of Drury on the positivity of a $$3\times 3$$ 3 × 3 matrix of the form $$(\Vert R_i^*R_j\Vert _{\mathop {\mathrm{tr}}\,})_{1 \le i, j \le 3}$$ ( ‖ R i ∗ R j ‖ tr ) 1 ≤ i , j ≤ 3 for any rectangular complex (or real) matrices $$R_1, R_2, R_3$$ R 1 , R 2 , R 3 so that the multiplication $$R_i^*R_j$$ R i ∗ R j is compatible for all $$i, j,$$ i , j , where $$\Vert \cdot \Vert _{\mathop {\mathrm{tr}}\,}$$ ‖ · ‖ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positivity of partitioned Hermitian matrices with unitarily invariant norms

, Volume 19 (3) – Sep 14, 2014
6 pages

/lp/springer_journal/positivity-of-partitioned-hermitian-matrices-with-unitarily-invariant-B0lHozElOK
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0306-4
Publisher site
See Article on Publisher Site

### Abstract

We give a short proof of a recent result of Drury on the positivity of a $$3\times 3$$ 3 × 3 matrix of the form $$(\Vert R_i^*R_j\Vert _{\mathop {\mathrm{tr}}\,})_{1 \le i, j \le 3}$$ ( ‖ R i ∗ R j ‖ tr ) 1 ≤ i , j ≤ 3 for any rectangular complex (or real) matrices $$R_1, R_2, R_3$$ R 1 , R 2 , R 3 so that the multiplication $$R_i^*R_j$$ R i ∗ R j is compatible for all $$i, j,$$ i , j , where $$\Vert \cdot \Vert _{\mathop {\mathrm{tr}}\,}$$ ‖ · ‖ 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.

### Journal

PositivitySpringer Journals

Published: Sep 14, 2014

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