Matrix equations, Index, G-Drazin inverses, Generalized inverses


Let $A$ and $E$ be $n \times n$ given complex matrices. This paper provides a necessary and sufficient condition for the solvability to the matrix equation system given by $AXA=AEA$ and $A^k E A X = X A E A^k$, for $k$ being the index of $A$. In addition, its general solution is derived in terms of a G-Drazin inverse of $A$. As consequences, new representations are obtained for the set of all G-Drazin inverses; some interesting applications are also derived to show the importance of the obtained formulas.

abs_vol35_pp503-510.pdf (141 kB)

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