# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 83: Limits of Sequences of Powers of Square Matrices

Let $A$ be a square matrix with complex entries, and let $S = \{ \lambda \in \mathbb{C} : | \lambda | < 1 \textrm{ or } \lambda = 1 \}$. Then $\lim _{n \rightarrow \infty} A^n$ exists if and only if both of the following hold:

• Every eigenvalue of $A$ is in $S$
• If $1$ is an eigenvalue of $A$, then its geometric multiplicity (the dimension of its eigenspace) equals its algebraic multiplicity.

Furthermore, the following are sufficient (but not necessary) conditions for the limit $\lim _{n \rightarrow \infty} A^n$ existing:

• Every eigenvalue of $A$ is in $S$ (as above)
• $A$ is diagonalizable

One application of this property is to the probability matrices of stochastic processes.