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MiS Preprint
49/2018

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

Paul Breiding, Khazhgali Kozhasov and Antonio Lerario

Abstract

We investigate some geometric properties of the real algebraic variety $\Delta$ 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 $\Delta$. We exhibit connections of our study to Real Algebraic Geometry (computing the Euclidean Distance Degree of $\Delta$) and Random Matrix Theory.

Received:
Jul 13, 2018
Published:
Jul 17, 2018
MSC Codes:
14P05, 15A22, 15A18
Keywords:
eigenvalues of real symmetric matrices, Euclidean distance degree

Related publications

inJournal
2018 Repository Open Access
Paul Breiding, Khazhgali Kozhasov and Antonio Lerario

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

In: Arnold mathematical journal, 4 (2018) 3-4, pp. 423-443