Abstract: It is well known that the set of all $n \times n$ symmetric matrices of rank $k$ is a smooth manifold. This set can be described as those symmetric matrices whose ordered vector of eigenvalues has exactly $n-k$ zeros. The set of all vectors in $\mathbb{R}^n$ with exactly $n-k$ zero entries is itself an analytic manifold.

In this work, we characterize the manifolds $M$ in $\mathbb{R}^n$ with the property that the set of all $n \times n$ symmetric matrices whose ordered vector of eigenvalues belongs to $M$ is a manifold. In particular, we show that if $M$ is a $C^k$ manifold then so is the corresponding matrix set for all $k \in \{2,3,\ldots, \infty, \omega\}$. We give a formula for the dimension of the matrix manifold in terms of the dimension of $M$.

This is a joint work with A. Daniilidis and J. Malick.