What does the PErron-Frobenius Theorem say, and why is it useful?

Perron-Frobenius theorem for irreducible matrices. Let A be an irreducible non-negative n * n matrix with period h and spectral radius ρ (A) = r. Then the following statements hold. The number r is a positive real number and it is an eigenvalue of the matrix A, called the Perron-Frobenius eigenvalue.