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

Paul Breiding; Khazhgali Kozhasov; Antonio Lerario

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arXiv · 1807.04530

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

Paul Breiding·Khazhgali Kozhasov·Antonio Lerario

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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.

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Paul Breiding, Khazhgali Kozhasov, Antonio Lerario. 2018-07-12. On the geometry of the set of symmetric matrices with repeated eigenvalues. https://arxiv.org/abs/1807.04530

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