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Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix?

Hope someone could answer these questions. Thanks in advance.

Rodrigo de Azevedo
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Heisenberg
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  • The inverse of any regular matrix is regular. – Martín-Blas Pérez Pinilla May 05 '14 at 09:50
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    It might be helpful to add the definitions of stochastic matrix and regular stochastic matrix to the question. – Juho Kokkala May 05 '14 at 10:19
  • @Martín-BlasPérezPinilla how about the counter example given by Juho below? – Heisenberg May 06 '14 at 05:29
  • @Heisenberg, I was thinking in regular=invertible. In Spanish is the most common meaning. – Martín-Blas Pérez Pinilla May 06 '14 at 06:12
  • In 2015, Reza Farhadian showed that there are some regular doubly stochastic matrices such that their inverses are regular doubly stochastic matrices [1]. Also, you can see [2, Appendix]. [1] http://neda.irstat.ir/article-1-229-fa.html. [2] Reza Farhadian, Nader Asadian F., On a New Class of Regular Doubly Stochastic Processes, American Journal of Theoretical and Applied Statistics. Vol. 6, No. 3, 2017, pp. 156-160. doi: 10.11648/j.ajtas.20170603.14 – user452330 Jun 14 '17 at 17:30

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Counterexample: consider the following stochastic matrix: \begin{equation} A = \begin{pmatrix} 2/3 & 1/3 \\ 1/3 & 2/3 \end{pmatrix}. \end{equation} Now, the inverse is \begin{equation} A^{-1} = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}, \end{equation} which is not even stochastic (entries are not nonnegative), let alone regular.

All powers of $A^{-1}$ also have negative nondiagonal entries, but this is not even needed as the definition of regular stochastic matrix requires stochasticity.

Juho Kokkala
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  • Thanks a lot. So there is no reason to study inverse of regular stochastic matrices right? – Heisenberg May 06 '14 at 05:26
  • Note that the row sums of the inverse of a right stochastic matrix must equal 1, and the column sums of the inverse of a left stochastic matrix must equal 1, because of http://math.stackexchange.com/questions/946776/show-that-sum-of-elements-of-rows-columns-of-a-matrix-is-equal-to-reciprocal-o – Simon Sep 25 '16 at 12:38