Solvability to some systems of matrix equations using G-outer inverses

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Dijana Mosić

Abstract

Two matrix equation systems $AXA=AEA$ and $BAEAX=XAEAD,$ where $A\in C^{m\times n}$ and $B, D, E\in C^{n\times m}$; and $AXA=AEA$, $BAEAX=B$ and $XAEAD=D$, where $A\in C^{m\times n}$, $B\in C^{p\times m}$, $D\in C^{n\times q}$ and $E\in C^{n\times m}$, are investigated and equivalent conditions for their solvability are presented. Expressions of their general solutions are established in terms of G-outer inverses of $A$. Specializing matrices $B, D, E$, these results are applied to solve various systems of matrix equations. In particular, the set of all G-outer inverses of $A$ is described. Since the fact $A$ is below $B$ under the G-outer $(T,S)$-partial order implies that any G-outer $(T,S)$-inverse of $B$ is also a G-outer $(T,S)$-inverse of $A$, an additional condition such that the converse holds is studied.

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