Perron–Frobenius theorem

The Perron-Frobenius theorem tells us that for an n × n matrix A with positive entries a_ij > 0 there is a positive real eigenvalue r of A such that any other eigenvalue λ satisfies |λ| < r. The bound r is referred to as the spectral radius of A.

(A matrix in which all entries are positive real numbers is here called positive and a matrix whose entries are non-negative real numbers is here called non-negative.)

Reference

• http://mathworld.wolfram.com/Perron-FrobeniusTheorem.html
• http://www.encyclopediaofmath.org/index.php/Perron-Frobenius_theorem
• http://en.wikipedia.org/wiki/Perron-Frobenius_theorem
• http://matrix.skku.ac.kr/sglee/perron_frobenius/perron_frobenius.html