  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.
