The largest real eigenvalue of a matrix is bigger than 1

Question

I have a problem which is interesting: given a real matrix $A_{n\times n}$, when this matrix has a largest real eigenvalue which is strictly bigger than 1. If possible, can you give some conditions that can guarantee this statement? Equivalently, this statement says that the entropy of the subshift of finite type is strictly bigger than 0.

Answer 1

The most famous theorem that I know of in this vein is the Perron-Frobenius Theorem. Here's a short summary: If all entries of a matrix are positive real numbers (or if the matrix is irreducible with non-negative real entries), then the spectral radius of the matrix is achieved by a positive real eigenvalue $r$. Further, $r$ is no smaller than the smallest row sum of $A$. Finally, $r$ can be efficiently found using some flavor of power iteration, since it is the eigenvalue corresponding to the unique eigenvector with all positive coordinates.