Positive matrices partitioned into a small number of Hermitian blocks

TL;DR

This paper explores how positive semidefinite matrices divided into a few Hermitian blocks can be expressed simply using their diagonal blocks, revealing a notable structural property.

Contribution

It introduces a straightforward method to represent such matrices based on the sum of their diagonal blocks, highlighting a new structural insight.

Findings

Positive matrices can be reconstructed from diagonal blocks

A simple representation formula is established

The property holds for matrices partitioned into a small number of blocks

Abstract

Positive semidefinite matrices partitioned into a small number of Hermitian blocks have a remarkable property. Such a matrix may be written in a simple way from the sum of its diagonal blocks