Geometric multiplicity margin for a submatrix (Q1611857)

Let \(A\) be a complex \(n\times n\) matrix. Denote the singular values of \(A\) by \(\sigma_{1}(A)\geq\sigma_{2}(A)\geq\dots\geq\sigma_{n}(A)\), and define \(i(A)\) to be the maximum of \(\dim\ker(\lambda I-A)\) taken over the eigenvalues \(\lambda\) of \(A.\) Consider the following problem: Given \(A\) and \(k>i(A)\), find an \(n\times n\) matrix \(Z\) which minimizes \(\|A-Z\|\) subject to the condition that \(i(Z)\geq k\) (where \(\|\;\|\) denotes the standard \(2\)-norm). \textit{J. M. Garcia, I. de Hoyos}, and \textit{F. E. Velasco} [Linear Multilinear Algebra 46, No. 1-2, 25-49 (1999; Zbl 0938.15005)] have shown that this minimum is equal to \(\min\sigma_{n-k+1}(\lambda I-A)\) taken over all \(\lambda\in\mathbb{C}\), and a minimizing choice of \(Z\) is described. The present paper solves the more difficult problem in which there is the added condition that \(Z\) is only allowed to differ from \(A\) in entries in a bottom right hand corner \(n_{2}\times n_{2}\) submatrix.