Structure of a nonnegative regular matrix and its generalized inverses
A matrix \(A\) is nonnegative (positive) if each entry of the matrix is nonnegative (positive). A nonnegative matrix \(A\) is called regular if it admits a nonnegative generalized inverse. In this paper, a complete description of all nonnegative generalized inverses of a nonnegative regular matrix \(A\) is given. In particular, the author proves the following important result: Let \(A\) be an \(m\times n\) regular matrix of rank \(r\) with no zero row or column, and let \(G\) be a nonnegative generalized inverse of \(A\). Then \(G\) is dominated by \(A'\), \(A'\) is the transpose of \(A\). Furthermore, there exist permutation matrices \(P\), \(Q\) such that \[ PAQ=\begin{bmatrix} \begin{array}{cccc|c} A_{11} & 0 & \cdots & 0 & {} \\ 0 & A_{22} & \cdots & 0 & \ast \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_{rr} & {} \\ \hline & & \ast & & \ast \end{array} \end{bmatrix}, \,Q^\prime GP^\prime= \begin{bmatrix} \begin{array}{cccc|c} G_{11} & 0 & \cdots & 0 & {} \\ 0 & G_{22} & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & G_{rr} & {} \\ \hline & & 0 & & \ast \end{array} \end{bmatrix} \] where \(A_{ii}\) is a positive, rank-one matrix and \(A_{ii}G_{ii} A_{ii}= A_{ii}\), \(i=1,2,\dots, r\).
scientific article; zbMATH DE number 1100485