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.