## Quick Reference

The theorem stating that a square matrix **A** with real non-negative elements has a positive real eigenvalue *λ*. The matrix is assumed to be irreducible, i.e. there is no permutation matrix **P** such that **PAP**′ has a zero submatrix in the bottom left-hand corner. Furthermore, every eigenvalue of **A** has modulus not exceeding *λ*, the eigenvalue *λ* is simple (i.e. *λ* is a non-multiple root of the characteristic equation (see matrix), and there is a corresponding eigenvector with positive elements. Perron originally established the theorem for matrices with positive elements, which are necessarily irreducible, and Frobenius extended the result.

*Subjects:*
Probability and Statistics.

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