Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C-Reference-Cited by-同舟云学术

Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C

Published:2021 Issue:1 Volume:11 Page:75
ISSN:2155-3289
Container-title:Numerical Algebra, Control & Optimization
language:
Short-container-title:NACO

Author:

Guerarra Sihem

Abstract

<p style='text-indent:20px;'>In this paper we derive the extremal ranks and inertias of the matrix <inline-formula><tex-math id="M1">\begin{document}$ X+X^{\ast}-P $\end{document}</tex-math></inline-formula>, with respect to <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ P\in\mathbb{C} _{H}^{n\times n} $\end{document}</tex-math></inline-formula> is given, <inline-formula><tex-math id="M4">\begin{document}$ X $\end{document}</tex-math></inline-formula> is a least rank solution to the matrix equation <inline-formula><tex-math id="M5">\begin{document}$ AXB = C $\end{document}</tex-math></inline-formula>, and then give necessary and sufficient conditions for <inline-formula><tex-math id="M6">\begin{document}$ X+X^{\ast}\succ P $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M7">\begin{document}$ \left( \geq P\text{, }\prec P\text{, }\leq P\right) $\end{document}</tex-math></inline-formula> in the Löwner partial ordering. As consequence, we establish necessary and sufficient conditions for the matrix equation <inline-formula><tex-math id="M8">\begin{document}$ AXB = C $\end{document}</tex-math></inline-formula> to have a Hermitian Re-positive or Re-negative definite solution.</p>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

Applied Mathematics,Control and Optimization,Algebra and Number Theory,Applied Mathematics,Control and Optimization,Algebra and Number Theory

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