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We investigate some geometric properties of the real algebraic variety$\Delta$ of symmetric matrices with repeated eigenvalues. We explicitly computethe volume of its intersection with the sphere and prove aEckart-Young-Mirsky-type theorem for the distance function from a genericmatrix to points in $\Delta$. We exhibit connections of our study to RealAlgebraic Geometry (computing the Euclidean Distance Degree of $\Delta$) andRandom Matrix Theory.

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