Dealing with the matrix of a simple difference equation $$\mathbf{x}^{(t+1)} = \mathbf{M} \mathbf{x}^{(t)}$$ it is often assumed the matrix, of dimension $n\times n$, admits $n$ distinct eigenvalues. In practice (programming), it seems that when the matrix $\mathbf{M}$ is nonsingular, and its entries are strictly positive, the above assumption is always verified. Do you agree with this empirical tendency? Is there some theoretical result that proves it rigorously?
Namely, let $\mathbf{M}$ be a real-valued nonsingular matrix of dimension $n\times n$. Does $\mathbf{M}$ admit $n$ distinct eigenvalues if we assume that the matrix entries are strictly greater than zero?
No, this is not true in general. Consider the non-singular matrix $$ A=\begin{pmatrix} 1 & 1 & 2\cr 7 & 1 & 1 \cr 2 & 1 & 1\end{pmatrix} $$ with only positive integer valued entries. It has the three eigenvalues $-1,-1,5$. They are not distinct. Such examples can be easily constructed more generally. (Of course one needs to say what you exactly mean by "empirical tendency".)