# 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

January 3, 2017

Posted by on Let be a square matrix with complex entries, and let . Then exists if and only if both of the following hold:

- Every eigenvalue of is in
- If is an eigenvalue of , 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 existing:

- Every eigenvalue of is in (as above)
- is diagonalizable

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

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