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Finding the closest normal structured matrix

Erna Begovic

TL;DR

This work addresses the problem of finding the closest normal matrix to a structured matrix, restricted to Hamiltonian, skew-Hamiltonian, per-Hermitian, and perskew-Hermitian classes, under the Frobenius norm. It recasts the problem as a dual maximization over unitary transformations and develops a structure-preserving Jacobi-type algorithm that iteratively enhances the diagonal energy via Givens-based rotations, producing a closest normal structured matrix X = Z diag(Z^H A Z) Z^H. The convergence analysis shows that accumulation points of the algorithm are stationary points of the diagonal-energy objective on the appropriate structure group, supported by gradient-based arguments and Polak’s framework. Numerical experiments validate the method, demonstrating fast energy concentration on the diagonal and robustness to pivot ordering, with applications to structured matrix proximity problems in related algebraic contexts.

Abstract

Given a structured matrix $A$ we study the problem of finding the closest normal matrix with the same structure. The structures of our interest are: Hamiltonian, skew-Hamiltonian, per-Hermitian, and perskew-Hermitian. We develop a structure-preserving Jacobi-type algorithm for finding the closest normal structured matrix and show that such algorithm converges to a stationary point of the objective function.

Finding the closest normal structured matrix

TL;DR

This work addresses the problem of finding the closest normal matrix to a structured matrix, restricted to Hamiltonian, skew-Hamiltonian, per-Hermitian, and perskew-Hermitian classes, under the Frobenius norm. It recasts the problem as a dual maximization over unitary transformations and develops a structure-preserving Jacobi-type algorithm that iteratively enhances the diagonal energy via Givens-based rotations, producing a closest normal structured matrix X = Z diag(Z^H A Z) Z^H. The convergence analysis shows that accumulation points of the algorithm are stationary points of the diagonal-energy objective on the appropriate structure group, supported by gradient-based arguments and Polak’s framework. Numerical experiments validate the method, demonstrating fast energy concentration on the diagonal and robustness to pivot ordering, with applications to structured matrix proximity problems in related algebraic contexts.

Abstract

Given a structured matrix we study the problem of finding the closest normal matrix with the same structure. The structures of our interest are: Hamiltonian, skew-Hamiltonian, per-Hermitian, and perskew-Hermitian. We develop a structure-preserving Jacobi-type algorithm for finding the closest normal structured matrix and show that such algorithm converges to a stationary point of the objective function.

Paper Structure

This paper contains 10 sections, 109 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Structured analogues of Theorem \ref{['tm:maximization-normal']}
  3. Hamiltonian and skew-Hamiltonian matrices
  4. Per-Hermitian and perskew-Hermitian matrices
  5. Jacobi-type algorithm for finding the closest normal matrix with a given structure
  6. Structure-preserving rotations
  7. Algorithm
  8. The choice of rotation angles $\phi$ and $\alpha$
  9. Convergence of Algorithm \ref{['agm:Jacobi-Ham_closest']}
  10. Numerical experiments

Figures (4)

  • Figure 1: Change in the absolute value of matrix entries.
  • Figure 2: Convergence of $\|\textnormal{diag}(A^{(k)})\|_F$, $A\in\mathcal{H}$.
  • Figure 3: Convergence of $\|\textnormal{diag}(A^{(k)})\|_F$, $A\in\mathcal{W}$.
  • Figure 4: Convergence of $\|\textnormal{diag}(A^{(k)})\|_F$, $A\in\mathcal{H}$, for two different pivot strategies.

Theorems & Definitions (11)

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  • ...and 1 more