Article Contents
Article Contents

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

• * Corresponding author: Sihem Guerarra
• 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.

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

 Citation:

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