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A regular matrix $A$ is described as a square matrix that for all positive integer $n$, is such that $A^n$ has positive entries.

How then would I prove something is regular? I mean I can prove something is irregular if $A^2$ has some 0 or negative entries; but I cant prove regularity since I cant solve $A^n$ for all integers $n$.

My thoughts are that if a matrix $A$ is diagonalisable as $A=PD^{-1}P$ then it is 'regular,' since then all $A^k$ exist; but does this also imply all entries of $A^k$ are positive?

Any hints?

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    The definition is not right. A regular matrix is a matrix for which *some* power of the matrix has all positive entries.2012-10-18
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    @ChristopherA.Wong Definitions can't be right or wrong.2013-02-27
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    yes but they can be consistent! According to wikipedia there is some ambiguity about "regular matrix", the top choice is the one given by @Christopher A. Wong. Though it probably would be helpful to specify that it is a "regular stochastic matrix".2017-03-05
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    @user53153 Definitions are useful if many people agree on them.2017-10-10

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