Ranks of Hermitian and skew-Hermitian solutions to the matrix equation \(AXA^*=B\)
Extremal ranks of solutions of some complex matrix equations and their applications are analyzed and discussed. The authors consider the following problems on solutions to the equation \(AXA^* = B\), where \(A \in {\mathbb C}^{m \times n}\) and \(B \in {\mathbb C}^{m \times m}\) are given with \(B^* = \pm B\) and \(X \in {\mathbb C}^{n \times n}\) is the unknown matrix: {\parindent=6mm \begin{itemize}\item[(i)] The maximal and minimal ranks of Hermitian and skew-Hermitian solutions \(X\), when \(B^* = B\) and \(B^* = -B\), respectively. \item[(ii)] The maximal and minimal ranks of the two real matrices \(X_0\) and \(X_1\) in Hermitian and skew-Hermitian solutions \(X = X_0 + iX_1\), when \(B^* = B\) and \(B^* = -B\), respectively. \end{itemize}} As applications, the maximal and minimal ranks of the real matrices \(C\) and \(D\) in a Hermitian (skew-Hermitian) \(g\)-inverse \((A + iB)^- = (C + iD)\) of a Hermitian (skew-Hermitian) matrix \(A + iB\) are shown.
scientific article; zbMATH DE number 5637031
scientific article