Equations of the form \(X\bar X = A\) with skew-Hamiltonian matrices \(A\)
type: Item
instance of: scholarly article
title: Equations of the form \(X\bar X = A\) with skew-Hamiltonian matrices \(A\)
zbMATH Open document ID: 1263.15034
DOI: 10.1134/S1064562412030271
published in: Doklady Mathematics
publication date: 2012-11-15T00:00:00Z
review text: For \(H = i \left( \begin{matrix} O_n & I_n\\ -I_n& O_n \end{matrix} \right) = i J\), a matrix \(B \in C^{2n,2n}\) is \(H\)-Hermitian if \(HB = B^*H\). Likewise \(A \in C^{2n,2n}\) is called skew-Hamiltonian if \(JA = A^TJ\). The paper proves that for each \(H\)-Hermitian matrix \(B\) the matrix \(A = B \overline{B}\) is skew-Hamiltonian and that for any skew-Hamiltonian matrix \(A\) that is similar to a real matrix without any real negative eigenvalues (phrased via the eigen- and Jordan structure of \(A\) as condition (a) in the paper) the equation \(X\overline{X} = A\) has an \(H\)-Hermitian matrix solution \(X\).
reviewed by: Frank Uhlig
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 6106469
zbMATH Keywords: matrix equation
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: skew-Hamiltonian matrix
zbMATH Keywords: eigenvalues
zbMATH Keywords: Jordan structure
author: Kh. D. Ikramov
MaRDI profile type: Publication
cites work: Antilinear operators and symplectic matrix algebra
full work available at URL: s1064562412030271
OpenAlex ID: W2058130390
description: scientific article; zbMATH DE number 6106469