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

Sihem Guerarra

doi:10.3934/naco.2020016

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2021, Volume 11, Issue 1: 75-86. Doi: 10.3934/naco.2020016

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

Sihem Guerarra^,

Faculty of Exact Sciences and Sciences of Nature and Life, Department of Mathematics and informatics, University of Oum El Bouaghi, 04000, Algeria

^* Corresponding author: Sihem Guerarra
^* Corresponding author: Sihem Guerarra

Received: May 31, 2019

Revised: August 31, 2019

Early access: February 2020

Published: March 2021

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Abstract

In this paper we derive the extremal ranks and inertias of the matrix $ X+X^{\ast}-P $, with respect to $ X $, where $ P\in\mathbb{C} _{H}^{n\times n} $ is given, $ X $ is a least rank solution to the matrix equation $ AXB = C $, and then give necessary and sufficient conditions for $ X+X^{\ast}\succ P $ $ \left( \geq P\text{, }\prec P\text{, }\leq P\right) $ in the Löwner partial ordering. As consequence, we establish necessary and sufficient conditions for the matrix equation $ AXB = C $ to have a Hermitian Re-positive or Re-negative definite solution.

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Mathematics Subject Classification: Primary: 15A24; Secondary: 15A09, 15A03, 15B57.

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