## 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.

### MSC:

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

  J. Ding and N. H. Rhee, When a matrix and its inverse are stochastic , The College Mathematics Journal, 44.2 , (2013), 108-109. · Zbl 06222742  J. Ding and A. Zhou, Nonnegative Matrices , Positive Operators, and Applications, World Scientific, 2009. · Zbl 1205.15048  C. Meyer, Matrix Analysis and Applied Linear Algebra , SIAM, Philadelphia, PA, 2000. · Zbl 0962.15001  R. Varga, Matrix Iterative Analysis , Prentice-Hall, Englewood Cliffs, NJ, 1962. · Zbl 0133.08602
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