Regular Matrix Definition

Question

I found a definition of Regular Matrices that is,

A regular matrix $A$ is a square matrix and there are some n ($\geq$1) such that all the entries of $A^n$ are positive.

I would like to know is this a correct definition? According to the definition, the following matrix is regular but how to prove it?

Any hints $\\ $\begin{pmatrix} 1 & 4& 3\\ 2& 0& 1\\ 4& 3& 2\\ \end{pmatrix}

Does this help? https://math.stackexchange.com/questions/216015/understanding-regular-matrices#:~:text=A%20regular%20matrix%20A%20is,An%20has%20positive%20entries. — Alessio K, Jun 11 '20 at 09:48
Yes I looked it up, but they didn`t discuss about the proof and also I am looking forward to the regular matrices without the stochastic explanation — Priya, Jun 11 '20 at 15:23
Can you add the reference that you have seen the definition? — narip, Jun 11 '22 at 14:05

score 3 · Answer 1 · answered Oct 14 '21 at 21:33

That definition (where we only use some power, and not all powers) is the usual definition for regularity of matrices, yes.

To verify your matrix is regular, you need only find some power $k$ whereby $A^k$ is full of only positive entries. For instance, $A^2$.

1 Answers1