Product of copies of a matrix and its transpose.

Question

Let $M$ be a square (entrywise) nonnegative matrix, and suppose that $M^n$ is (entrywise) positive for some $n\in\mathbb{N}$. Consider now a product of $n$ matrices such that each of the matrices appearing in the product is either $M$ or the transpose of $M$. Is the product (entrywise) positive? If not, is it always possible to find an $n'$ for which the product of $n'$ factors as above is always (entrywise) positive?

Answer 1

I played around with it a bit after misreading your question. If you take

$$M = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ \end{pmatrix}$$

then $M^2$ is positive in all entries but if you take $M\cdot M^T$ it is not. Not sure if that's what you have in mind but if I didn't misread your question again that answers it in the negative.

Edit to answer the comment:

Take $M=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 0 \\ \end{pmatrix}$.

Then $M^5$ is entrywise positive but $M\cdot M^T\cdot M^3$ has zeros and so has $M\cdot M^T\cdot M^2,M\cdot M^T\cdot M,M\cdot M^T$.