Understanding matrices whose powers have positive entries

Question

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?

Answer 1

If $A$ has an entry that is $0$ or negative, then $A$ is not regular. If, on the other hand, every entry in $A$ is positive, can $A^2$ have a negative or zero entry? Can $A^3$? There’s an easy proof by induction waiting here for you to find it. Note that diagonalizability has nothing to do with the matter: if $A$ is square, $A^n$ exists for all $n\ge 0$ whether or not $A$ is diagonalizable. Diagonalizability of $A$ merely makes it easy to calculate the powers of $A$.

However, that’s not the usual definition of regular matrix. The usual definition is that a square matrix $A$ is regular if it is stochastic and there is some $n\ge 1$ such that all of the entries of $A^n$ are positive.

Answer 2

Linear combinations of positive numbers (with positive coefficients) will always yield positive numbers, yes? Therefore, it should suffice to note that all entries of $A$ are positive. On the other hand, if $A$ has an entry that is nonpositive, then $A$ can't be regular (since $A=A^1$ and $1>0$), so this is a sufficient condition, too.