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

Ding, J.; Rhee, N. H.

Missouri J. Math. Sci. 26, No. 1, 98-103 (2014).

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.

Reviewer: Ali Reza Moghaddamfar (Tehran)

MSC:

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

Keywords:

stochastic matrix; permutation matrix; nonnegative matrix; canonical form

References:

[1]	J. Ding and N. H. Rhee, When a matrix and its inverse are stochastic , The College Mathematics Journal, 44.2 , (2013), 108-109. · doi:10.4169/college.math.j.44.2.108
[2]	J. Ding and A. Zhou, Nonnegative Matrices , Positive Operators, and Applications, World Scientific, 2009. · Zbl 1205.15048
[3]	C. Meyer, Matrix Analysis and Applied Linear Algebra , SIAM, Philadelphia, PA, 2000. · Zbl 0962.15001
[4]	R. Varga, Matrix Iterative Analysis , Prentice-Hall, Englewood Cliffs, NJ, 1962. · Zbl 0133.08602