Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices

TL;DR

This paper explores inequalities involving positive semi-definite block matrices, extending known results by analyzing symmetric norm bounds and providing new insights into their structural properties.

Contribution

The paper extends the class of positive semi-definite block matrices satisfying symmetric norm inequalities, based on a decomposition lemma, advancing understanding of matrix inequalities.

Findings

Established new symmetric norm inequalities for block matrices.

Extended the class of matrices satisfying these inequalities.

Provided theoretical insights into matrix decomposition and norm bounds.

Abstract

For positive semi-definite block-matrix $M,$ we say that $M$ is P.S.D. and we write $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix} \in {\mathbb{M}}\_{n+m}^+$, with $A\in {\mathbb{M}}\_n^+$, $B \in {\mathbb{M}}\_m^+.$ The focus is on studying the consequences of a decomposition lemma due to C.~Bourrin and the main result is extending the class of P.S.D. matrices $M$ written by blocks of same size that satisfies the inequality: $\|M\|\le \|A+B\|$ for all symmetric norms.