Perron-Frobenius Theorem

An \( n \times n \) matrix \( A \) is said to be irreducible if for all pairs \( (i, j) \) there exists some \( k \in \mathbb{Z} \) such that \( M_{i, j}^k > 0 \).

We say an \( n \times n \) matrix \( A \) is primitive if there exists some \( k \) for which all entries of \( A^k \) are positive. We can see that primitivity is a similar yet stronger condition than irreducibility.

If \( A \) is an irreducible matrix then \( A \) has an eigenvalue \( \lambda_0 \) satisfying:

1. All other eigenvalues of \( A \) satisfy \( |\lambda| < \lambda_0 \)
2. \( \lambda_0 \) has algebraic and geometric multiplicity equal to one
3. \( \lambda_0 \) has an eigenvector with all entries greater than zero
4. Further, any non-negative eigenvector of \( A \) is a multiple of this eigenvector