Prescribed Eigenvalues via Optimal Perturbation of main-diagonal submatrix

Abstract

Consider a given square matrix $\text{K}$ with square blocks $A_{11},A_{22},\dots ,A_{nn}$ on the main diagonal. This paper aims to compute an optimal perturbation $\Delta$ of a preassigned block $A_{ii}\in {\mathbb{C}}^{d_i\times d_k},\left(1\le i\le n\right)$,with respect to the spectral norm distance, such that the perturbed matrix ${\text{K}}_X$ has $k\le d_i$ prescribed eigenvalues. This paper presents a method for constructing the optimal perturbation by improving and extending the methodology, necessary definitions and lemmas of previous related works. Some conceivable applications of this subject are also presented. Numerical experiments are provided to illustrate the validity of the method.