An algebraic characterization of non-singular matrix semicircles

TL;DR

This paper characterizes when a matrix semicircle is non-singular at zero through algebraic and analytic conditions, linking matrix pencil decomposability, covariance map properties, and spectral density.

Contribution

It establishes new equivalences connecting algebraic decomposability, covariance map scalability, and spectral properties of matrix semicircles, using advanced algebraic and analytic techniques.

Findings

Equivalence between matrix pencil decomposability and covariance map scalability.

Spectral density at zero is explicitly related to the trace of a minimal solution.

Removal of a stability hypothesis from previous bifurcation analyses.

Abstract

Let $A_1, \ldots, A_r$ be Hermitian $n \times n$ matrices and $S = \sum A_i \otimes s_i$ the associated matrix semicircle, where $s_1, \ldots, s_r$ are free semicircular variables. We prove that the following are equivalent: (i) the matrix pencil $A = \sum A_i x_i$ is LR-semisimple (decomposes, up to left--right equivalence, as a direct sum of unsplittable pencils); (ii) $S$ is non-singular at $t = 0$ (the matrix-valued Cauchy transform has a continuous boundary limit near the origin); (iii) the covariance map $\eta\colon X \mapsto \sum A_i X A_i$ is symmetrically DS-scalable (there exists $C \succ 0$ with $\eta(C) = C^{-1}$). When these hold, the spectral density satisfies $f(0) = \frac{1}{\pi}\,\mathrm{tr}(C)$, where $C$ is the unique trace minimizer of the solution set $\{W \succ 0 : \eta(W)\,W = I\}$. The proof combines algebraic and analytic ingredients. On the algebraic side, we establish the equivalence (i) $\Leftrightarrow$ (iii) using Gurvits' capacity theory for indecomposable maps and a geodesic reflection theorem in the Riemannian manifold of positive definite matrices, which upgrades DS-scalability to symmetric DS-scalability for self-adjoint completely positive maps. On the analytic side, we prove (iii) $\Rightarrow$ (ii) via a Lyapunov--Schmidt reduction of Speicher's equation at a trace-minimizing solution, showing that the Jacobian of the bifurcation equations is positive definite. This removes a stability hypothesis that was required in earlier approaches.