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Prescribed Eigenvalues via Optimal Perturbation of main-diagonal submatrix

M. R. Eslahchi, E. Kokabifar

TL;DR

The paper addresses updating a structured matrix by perturbing a single main-diagonal block to enforce a prescribed spectral subset $\Lambda$ while minimizing the spectral-norm distance. It introduces a structured singular-value framework based on $Q_T(\gamma)$ and $\mathcal{S}_k(D,\gamma)$, and then derives optimal perturbations via $\\Delta_* = - \alpha^* U(\gamma_*) V(\gamma_*)^{\dag}$, guaranteeing that $\Lambda$ appears in the spectrum of the perturbed matrix. The authors extend the method from the 3-eigenvalue case to general $k\le m$, provide a permutation-based extension to perturb any diagonal block, and illustrate applications in dynamic system stability and spectral clustering with numerical validation. These results yield a provably optimal, computationally implementable approach for structured matrix nearness problems with prescribed spectral requirements, with potential impact in control, model updating, and data-driven clustering.

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 $Δ$ 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.

Prescribed Eigenvalues via Optimal Perturbation of main-diagonal submatrix

TL;DR

The paper addresses updating a structured matrix by perturbing a single main-diagonal block to enforce a prescribed spectral subset while minimizing the spectral-norm distance. It introduces a structured singular-value framework based on and , and then derives optimal perturbations via , guaranteeing that appears in the spectrum of the perturbed matrix. The authors extend the method from the 3-eigenvalue case to general , provide a permutation-based extension to perturb any diagonal block, and illustrate applications in dynamic system stability and spectral clustering with numerical validation. These results yield a provably optimal, computationally implementable approach for structured matrix nearness problems with prescribed spectral requirements, with potential impact in control, model updating, and data-driven clustering.

Abstract

Consider a given square matrix with square blocks on the main diagonal. This paper aims to compute an optimal perturbation of a preassigned block ,with respect to the spectral norm distance, such that the perturbed matrix has 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.
Paper Structure (9 sections, 9 theorems, 68 equations, 2 figures)

This paper contains 9 sections, 9 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Nearest submatrix providing three prescribed eigenvalues
  3. Computing lower bounds for the optimal perturbation
  4. Construction of the optimal perturbation
  5. $k$ Prescribed eigenvalues
  6. Lower bounds for the optimal perturbation
  7. Construction of the optimal perturbation
  8. Making optimal change to every main diagonal block to have prescribed eigenvalues
  9. Conceivable application and numerical examples

Key Result

Corollary 2.1

Let $\lambda _1 ,\lambda _2 , \lambda _3$ be eigenvalues of $T \in {\mathbb{C}^{d \times d}}$. Then for all $\gamma$ we have

Figures (2)

  • Figure 1: Graphs of ${s_{k\left( {m - 1} \right) + 1}}\left( {{\mathcal{S}_k}\left( {D,\gamma_* } \right)} \right)$ (the black graph) and ${s_{k\left( {m - 1} \right)}}\left( {{\mathcal{S}_k}\left( {D,\gamma_* } \right)} \right)$(the red graph) when K is a normal matrix.
  • Figure 2: The graph of the function $s_{10}\left(\gamma\right)$.

Theorems & Definitions (11)

  • Corollary 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • Corollary 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 1 more