# My matrices are non-negative,stochastic, irreducible and aperiodic, I want to know whether they always converge in power iteration.

I am working on a problem of SCC graph. The matrix representation of graph will be a square non-negative matrix that is column stochastic, irreducible. I will make it aperiodic by adding a self-loop on each node so that I can use the power method.
I will write a program of power method. Now I want to know whether my matrix always converges.
I know that diagonalizable matrix always converges. But if my matrix is not diagonalizable for some graph then will it converge? So what should I do if I want my matrix to converge always? Or my matrix's property(non-negative, stochastic, irreducible, aperiodic) is enough to converge always?
(My program will take input any SCC graph, I will add self-loop on each node. And write a program of power method. )

• what does it mean for a matrix to converge? – Dan Rust Sep 14 '19 at 10:56
• We can get the dominant eigenvalue of a square matrix if it is Diagonalizable and has s dominant eigenvalue by power method. If the matrix has a dominant eigenvalue then in power method the initial vector converges to a dominant eigenvector value. @DanRust – criticalmind Sep 14 '19 at 11:17

Your matrices are primitive which means that every entry is a positive real number. In particular, it means that we may apply the Perron-Frobenius theorem, one consequence of which is that if $$M$$ is a primitive matrix with (dominant) PF-eigenvalue $$\lambda$$, and associated left- and right-PF eigenvectors $$\textbf{l},\textbf{r}$$, respectively (normalised so that $$\mathbf{l}^T\mathbf{r} = 1$$), and we have $$\lim_{n \to \infty}M^n/\lambda^n = \mathbf{r}\mathbf{l}^T.$$