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On normal and structured matrices under unitary structure-preserving transformations

Erna Begovic, Heike Fassbender, Philip Saltenberger

TL;DR

Structured canonical forms under unitary and suitable structure-preserving similarity transformations for normal and (skew-)Hamiltonian as well as normal and per(skeW)-Hermitian matrices are proposed.

Abstract

Structured canonical forms under unitary and suitable structure-preserving similarity transformations for normal and (skew-)Hamiltonian as well as normal and per(skew)-Hermitian matrices are proposed. Moreover, an algorithm for computing those canonical forms is sketched.

On normal and structured matrices under unitary structure-preserving transformations

TL;DR

Structured canonical forms under unitary and suitable structure-preserving similarity transformations for normal and (skew-)Hamiltonian as well as normal and per(skeW)-Hermitian matrices are proposed.

Abstract

Structured canonical forms under unitary and suitable structure-preserving similarity transformations for normal and (skew-)Hamiltonian as well as normal and per(skew)-Hermitian matrices are proposed. Moreover, an algorithm for computing those canonical forms is sketched.

Paper Structure

This paper contains 11 sections, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structured canonical form for normal (skew-)Hamiltonian matrices under unitary symplectic similarity transformation
  3. Canonical form for normal Hamiltonian matrices
  4. Canonical form for normal skew-Hamiltonian matrices
  5. Structured canonical form for normal per(skew)-Hermitian matrices under unitary perplectic similarity transformation
  6. Canonical form for normal per-Hermitian matrices
  7. Canonical form for normal perskew-Hermitian matrices under unitary perplectic similarity transformation
  8. Algorithm for computing the canonical form of normal (skew-)Hamiltonian matrices
  9. Convergence of the proposed algorithm
  10. Numerical Experiments
  11. Concluding Remarks

Figures (2)

  • Figure 1: Nonzero pattern of the skew-Hermitian part of a $30 \times 30$ normal Hamiltonian matrix $H$ which can be diagonalized by a unitary symplectic transformation, structure-preserving algorithm on the left, unstructured algorithm on the right.
  • Figure 2: Nonzero pattern of the skew-Hermitian part of a $30 \times 30$ normal Hamiltonian matrix $H$ with purely imaginary eigenvalues, structure-preserving algorithm on the left, unstructured algorithm on the right.