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

We investigate some geometric properties of the real algebraic variety \u394 of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart\u2013Young\u2013Mirsky-type theorem for the distance function from a generic matrix to points in \u394. We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of \u394) and random matrix theory