On the geometry of the set of symmetric matrices with repeated eigenvalues

Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario

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Submission date: 13. Jul. 2018
Pages: 16
published in: Arnold mathematical journal, 4 (2018) 3-4, p. 423-443 
DOI number (of the published article): 10.1007/s40598-018-0095-0
Bibtex
MSC-Numbers: 14P05, 15A22, 15A18
Keywords and phrases: eigenvalues of real symmetric matrices, Euclidean distance degree
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Abstract:
We investigate some geometric properties of the real algebraic variety Δ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in Δ. We exhibit connections of our study to Real Algebraic Geometry (computing the Euclidean Distance Degree of Δ) and Random Matrix Theory.