Group inverses of \(M\)-matrices associated with nonnegative matrices having few eigenvalues
Several contributions to the study of the group generalized inverse \(A^\#\) of singular irreducible \(M\)-matrices \(A = rI - M\) are given in this note. The first results are explicit formulas for \(A^\#\), if \(M\) is of low rank. They allow to study when \(A^\#\) is an \(M\)-matrix, too. These results are applied to two interesting classes of matrices \(M\), namely doubly regular tournament matrices and magic squares of order \(4k\), as they are produced by MATLAB. Their eigenvalues are determined, and as their ranks are low, the group inverse can be explicitly given. Also their singular values are exhibited.
scientific article; zbMATH DE number 769026
scientific article