Common nonnegative definite solutions of some classical matrix equations. (English) Zbl 1391.15060
Let \(\mathbb{C}^{m\times n}\) and \(\mathbb{C}_{\geq}^{m}\) stand for the set of all \(m\times n\) complex matrices and the set of all \(m\times m\) Hermitian nonnegative definite matrices, respectively, and let \(A^{*}\) stand for the conjugate transpose of a matrix \(A\in \mathbb{C}^{m\times n}\). In this paper, the authors examine a necessary and sufficient condition for the existence of a Hermitian nonnegative definite solution of the system:
\[
A_{1}X=C_{1}, \;XB_{2}=C_{2},\;A_{3}X{A_{3}}^{*}=C_{3}, \;A_{4}X{A_{4}}^{*}=C_{4},
\]
where \(A_{1},C_{1}\in\mathbb{C}^{m\times n}\), \(B_{2},C_{2}\in\mathbb{C}^{n\times s}\), \(A_{3},A_{4}\in \mathbb{C}_{\geq}^{n}\) and \(C_{3},C_4\in\mathbb{C}^{n\times n}\). Moreover, the authors give some special cases of their main results. Furthermore, they give a numerical example to illustrate the results of the paper.
Reviewer: Süleyman Güler (Aydin)
MSC:
| 15A24 | Matrix equations and identities |
| 65F30 | Other matrix algorithms (MSC2010) |
| 15A03 | Vector spaces, linear dependence, rank, lineability |
| 15A09 | Theory of matrix inversion and generalized inverses |
Keywords:
Moore-Penrose inverse; matrix equation; Hermitian nonnegative definite solution; numerical exampleReferences:
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