Nonnegative rectangular matrices having certain nonnegative 𝑊-weighted group inverses

Jain, S. K.

doi:10.1090/S0002-9939-1982-0647886-2

Nonnegative rectangular matrices having certain nonnegative $W$-weighted group inverses
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Proc. Amer. Math. Soc. 85 (1982), 1-9 Request permission

Abstract:

Nonnegative rectangular matrices having nonnegative $W$-weighted group inverses are characterized. Our techniques suggest an interesting approach to extend the earlier known results on $\lambda$-monotone square matrices to rectangular ones. We also answer a question of characterizing nonnegative matrices having a nonnegative solution $X$ where (1) $A = AXA$, (2) $X = XAX$, (3) $(AX)$ is $0$-symmetric, (4) $(XA)$ is $0$-symmetric. In particular, we obtain theorems of Berman-Plemmons and Plemmons-Cline characterizing nonnegative matrices $A$ with a nonnegative Moore-Penrose inverse. Matrices having nonnegative generalized inverses are of interest in the study of finding nonnegative best approximate solutions of linear systems. Such matrices are of considerable interest in statistics, numerical linear algebra and mathematical economics.

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