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.

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