Some of the things I've learned every day since Oct 10, 2016
83: Limits of Sequences of Powers of Square Matrices
January 3, 2017Posted 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.