On the Geometry of the Set of Symmetric Matrices with Repeated Eigenvalues

Source abstract

Abstract We investigate some geometric properties of the real algebraic variety $$\Delta$$ 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 $$\Delta$$ Δ . We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of $$\Delta$$ Δ ) and random matrix theory.