Prescribed Eigenvalues via Optimal Perturbation of main-diagonal submatrix

TL;DR

This paper develops a method to compute optimal perturbations of a block within a matrix to achieve prescribed eigenvalues, extending previous methods and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces an improved method for optimal block perturbation to assign specific eigenvalues, expanding on existing theoretical frameworks.

Findings

The method successfully computes perturbations for prescribed eigenvalues.

Numerical experiments confirm the method's validity and effectiveness.

Applications of the perturbation technique are demonstrated.

Abstract

Consider a given square matrix $\textrm {K}$ with square blocks $A_{11},A_{22},\ldots,A_{nn}$ 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 ${\textrm {K}_X}$ 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.