Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
On the geometry of the set of symmetric matrices with repeated eigenvalues
Paul Breiding, Khazhgali Kozhasov and Antonio Lerario
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.