Random spectrahedra
Paul Breiding, Khazhgali Kozhasov, Antonio Lerario

TL;DR
This paper studies the geometric and probabilistic properties of random spectrahedra, focusing on the expected number of singular points and their relation to eigenvalue multiplicities, with explicit formulas for specific cases.
Contribution
It introduces a probabilistic framework for analyzing random spectrahedra, deriving explicit formulas for expected singular points and connecting these to eigenvalue multiplicity volumes.
Findings
Expected singular points on 3D spectrahedra relate to eigenvalue coincidence volume.
Explicit formula for quartic spectrahedra: E[σ_4] = 6 - 4/√3.
Average singular points on the variety of singular matrices: n(n-1).
Abstract
Spectrahedra are affine-linear sections of the cone of positive semidefinite symmetric -matrices. We consider random spectrahedra that are obtained by intersecting~ with the affine-linear space , where is the identity matrix and is an -dimensional linear space that is chosen from the unique orthogonally invariant probability measure on the Grassmanian of -planes in the space of real symmetric matrices (endowed with the Frobenius inner product). Motivated by applications, for we relate the average number of singular points on the boundary of a three-dimensional spectrahedron to the volume of the set of symmetric matrices whose two smallest eigenvalues coincide. In the case of quartic spectrahedra () we show that $\mathbb{E} \sigma_4 =…
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Taxonomy
TopicsCultural Heritage Materials Analysis · Geochemistry and Geologic Mapping · Electron and X-Ray Spectroscopy Techniques
