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**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.

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 |

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\textit{J. Ding} and \textit{N. H. Rhee}, Missouri J. Math. Sci. 26, No. 1, 98--103 (2014; Zbl 1339.15022)

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### 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. · Zbl 06222742 |

[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 |

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