Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Assume $A$ is diagonalizable over $\mathbb{C}$. Show that, if $\rho(A) < 1$ then $\lim_{k \rightarrow +\infty} A^k = 0$.
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Assume $A$ is diagonalizable over $\mathbb{C}$. Show that, if $\rho(A) < 1$ then $\lim_{k \rightarrow +\infty} A^k = 0$.