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Procrustes problem for the inverse eigenvalue problem of normal (skew) $J$-Hamiltonian matrices and normal $J$-symplectic matrices

S. Gigola, L. Lebtahi, N. Thome

Abstract

A square complex matrix $A$ is called (skew) $J$-Hamiltonian if $AJ$ is (skew) hermitian where $J$ is a real normal matrix such that $J^2=-I$, where $I$ is the identity matrix. In this paper, we solve the Procrustes problem to find normal (skew) $J$-Hamiltonian solutions for the inverse eigenvalue problem. In addition, a similar problem is investigated for normal $J$-symplectic matrices.

Procrustes problem for the inverse eigenvalue problem of normal (skew) $J$-Hamiltonian matrices and normal $J$-symplectic matrices

Abstract

A square complex matrix is called (skew) -Hamiltonian if is (skew) hermitian where is a real normal matrix such that , where is the identity matrix. In this paper, we solve the Procrustes problem to find normal (skew) -Hamiltonian solutions for the inverse eigenvalue problem. In addition, a similar problem is investigated for normal -symplectic matrices.
Paper Structure (4 sections, 8 theorems, 118 equations)

This paper contains 4 sections, 8 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Optimization problem for normal $J$-Hamiltonian matrices
  3. The normal skew $J$-Hamiltonian case
  4. The normal $J$-symplectic case

Key Result

Lemma 1

Let $B \in {\mathbb C}^{q \times m}$, $P_1 \in {\mathbb C}^{q \times q}$, and $P_2 \in {\mathbb C}^{m \times m}$ where $P_i^2=P_i=P_i^*$ for $i=1,2$. Then the following statements hold:

Theorems & Definitions (16)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Remark 2
  • Remark 3
  • Example 1
  • Example 2
  • Theorem 2
  • Theorem 3
  • ...and 6 more