A formula for the 2-norm distance from a matrix to the set of matrices with multiple eigenvalues

Summary.

We prove that the 2-norm distance from an $n\times n$ matrix A to the matrices that have a multiple eigenvalue $\lambda$ is equal to \[ rsep_{\lambda}(A)=\max_{\gamma\ge 0}\sigma_{2n-1}\left(\begin{array}{cc} A-\lambda I & \gamma I 0 & A-\lambda I \end{array}\right), \] where the singular values $\sigma_{k}$ are ordered nonincreasingly. Therefore, the 2-norm distance from A to the set of matrices with multiple eigenvalues is \[ rsep(A)=\min_{\lambda\in\mathbb{C}}rsep_{\lambda}(A). \]