{1}-inverses of Square Matrices and Rational Canonical Form
Mathematica Moravica, Tome 1 (1997) no. 1.

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In this paper we solve the first Penrose’s equation $AXA = A$ for square real matrices A using the rational canonical form of matrices. The idea is to find ${1}$-inverse $X$ of $A$ using similarity $X = TZT^{−1}$, where $Z$ is ${1}$-inverse of $B$ and $A = TBT^{−1}$ is the rational canonical representation of $A$. 
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     author = {Ljubi\v{s}a Ko\v{c}inac and Predrag Stanimirovi\'c and Slavi\v{s}a {\DJ}or{\dj}evi\'c},
     title = {{1}-inverses of {Square} {Matrices} and {Rational} {Canonical} {Form}},
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Ljubiša Kočinac; Predrag Stanimirović; Slaviša Đorđević. {1}-inverses of Square Matrices and Rational Canonical Form. Mathematica Moravica, Tome 1 (1997) no. 1. https://geodesic-test.mathdoc.fr/item/MM3_1997_1_1_a7/