Moment of a subspace and joint numerical range

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 $W$.