Moment of a subspace and joint numerical range

TL;DR

This paper introduces the concept of the moment of a subspace in complex vector spaces, explores its geometric properties, and links it to joint numerical ranges to facilitate the analysis of minimal hermitian matrices.

Contribution

It provides a new geometric and algebraic framework for understanding moments of subspaces and their relation to joint numerical ranges, aiding in the characterization of minimal matrices.

Findings

Characterization of extremal points and curves of the moment set

Relation of the moment set to joint numerical ranges of rank-one matrices

A new method for constructing or detecting minimal matrices

Abstract

For a given complex finite dimensional subspace $S$ of $\mathbb{C}^n$ and a fixed basis, we study the compact and convex subset of $\left(\mathbb{R}_{\geq 0}\right)^n$ that we call the moment of $S$ $m_S=$ convex hull ($\{|s|^2\in\mathbb{R}^n_{\geq 0}: s\in S \wedge \|s\|=1\} )$ $\simeq \{ Diag(Y) \in M_n^h(\mathbb{C}):Y\geq 0, tr(Y)=1, P_S Y P_S=Y\}$ where $|s|^2=(|s_1|^2,|s_2|^2,\dots,|s_n|^2)$. This set is relevant in the determination of minimal hermitian matrices ($M\in M^h_n$ such that $\|M+D\|\leq D$ for every diagonal $D$ and $\| \|$ the spectral norm). We describe extremal points and curves of $m_S$ in terms of principal vectors that minimize the angle between $S$ and the coordinate axes. We also relate $m_S$ to the joint numerical range $W$ of $n$ rank one $n\times n$ matrices constructed with the orthogonal projection $P_S$ and the fixed basis used. This connection provides a new approach to the description of $m_S$ and to minimal matrices. As a consequence the intersection of two of these joint numerical ranges allow the construction or detection of a minimal matrix, a fact that is easier to corroborate than the equivalent condition for moments. It is also proved that $m_S$ is a semi-algebraic set equal to the intersection of the mentioned $W$ with a hyperplane and whose generated positive cone coincides with that of $W$.