# Supports for minimal hermitian matrices

## TL;DR

This paper investigates the structure of supports related to minimal Hermitian matrices, introducing an invariant to measure support proximity and analyzing the geometric properties of the support set within flag manifolds.

## Contribution

It defines a new invariant for supports of minimal Hermitian matrices and studies their geometric structure using critical point analysis.

## Key findings

- The invariant δ effectively measures how close subspace pairs are to forming supports.
- Supports have interior points in the space of flag manifolds, indicating their richness.
- Analysis of the map F provides insights into the structure of supports.

## Abstract

We study certain pairs of subspaces $V$ and $W$ of $\mathbb{C}^n$ we call supports that consist of eigenspaces of the eigenvalues $\pm\|M\|$ of a minimal hermitian matrix $M$ ($\|M\|\leq \|M+D\|$ for all real diagonals $D$). For any pair of orthogonal subspaces we define a non negative invariant $\delta$ called the adequacy to measure how close they are to form a support and to detect one. This function $\delta$ is the minimum of another map $F$ defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of $F$ in order to approximate $\delta$. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.