Nearest southeast submatrix that makes multiple a prescribed eigenvalue. (Q999800)

Let \(A\in \mathbb C^{n\times n}\), \(B\in \mathbb C^{n\times m}\), \(C\in \mathbb C^{m\times n}\), \(D\in \mathbb C^{m\times m}\) be given complex matrices and \(z_0\in \mathbb C\). Let \(\mathcal X\) be the set of all \(X\in \mathbb C^{m\times m}\) such that \(z_0\) is a multiple eigenvalue of \(\left(\begin{smallmatrix} A&B\\ C&X\end{smallmatrix}\right)\). The question: what is \(\text{dist}(D,\mathcal X)\) is investigated. The problem is first reduced to the case \(z_0=0\). It is proved that if \(z_0=0\) is not an eigenvalue of \(A\), then \(\text{dist}(D,\mathcal X)=\sup_{t\in\mathbb R}\sigma_{2m-1}(S_2(t,D))\) where \(S_2(t,D)\) is a \(2m\times2m\) matrix (its construction is too complicated to be stated here) and \(\sigma_{2m-1}(S_2(t,D))\) denotes its \((2m-1)\)th (in decreasing order) singular value. In the case when \(z_0\) is an eigenvalue of \(A\) an interesting conjecture is stated.