Nonnegative integral matrices having generalized inverses
Commun. Korean Math. Soc. 2014 Vol. 29, No. 2, 227-237
https://doi.org/10.4134/CKMS.2014.29.2.227
Printed April 1, 2014
Kyung-Tae Kang, LeRoy B. Beasley, Luis Hernandez Encinas, and Seok-Zun Song
Jeju National University, Utah State University, Spanish National Research Council, Jeju National University
Abstract : For an $m\times n$ nonnegative integral matrix $A$, a {\it generalized inverse} of $A$ is an $n\times m$ nonnegative integral matrix $G$ satisfying $AGA=A$. In this paper, we characterize nonnegative integral matrices having generalized inverses using the structure of nonnegative integral idempotent matrices. We also define a {\it space decomposition} of a nonnegative integral matrix, and prove that a nonnegative integral matrix has a generalized inverse if and only if it has a space decomposition. Using this decomposition, we characterize nonnegative integral matrices having reflexive generalized inverses. And we obtain conditions to have various types of generalized inverses.
Keywords : idempotent matrix, regular matrix, generalized inverse matrix
MSC numbers : Primary 15A09, 15A23, 15B36
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