When a matrix and its inverse are nonnegative. (English) Zbl 1339.15022

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; doi:10.4169/college.math.j.44.2.108)].
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.


15B48 Positive matrices and their generalizations; cones of matrices
15B51 Stochastic matrices
15A21 Canonical forms, reductions, classification
Full Text: Euclid


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