When a matrix and its inverse are nonnegative
A matrix is called stochastic if it is a nonnegative matrix for which each of its row sums equals \(1\). It is clear that if \(A\) is a permutation matrix, then \(A\) and \(A^{-1}\) are stochastic. The converse of this statement is also true and its proof has been given by the authors in [``Teaching tip: when a matrix and its inverse are stochastic'', Coll. Math. J. 44, No. 2, 108--109 (2013; \url{doi:10.4169/college.math.j.44.2.108})].NEWLINENEWLINE In this article, they present another proof of this fact based on the canonical forms of stochastic matrices. They also extend this result to show that that \(A\) and \(A^{-1}\) are nonnegative if and only if \(A\) is a product of a diagonal matrix with all positive diagonal entries and a permutation matrix.
scientific article; zbMATH DE number 6359510