A CONTRIBUTION TO COLLATZ’S EIGENVALUE INCLUSION THEOREM FOR NONNEGATIVE IRREDUCIBLE MATRICES ∗

Abstract. The matrix calculus is widely applied in various branches of mathematics and control system engineering. In this paper properties of real matrices with nonnegative elements are studied. The classical Collatz theorem is unique and immediately applicable to estimatingthe spectral radius of nonnegative irreducible matrices. The coherence property is identified. Then the Perron–Frobenius theorem and Collatz’s theorem are used to formulate the coherence property more precisely. It is shown how dual variation principles can be used for the iterative calculation of x = X[A] andthe spectral radius of A, wherex is any positive n-vector, X[A] is the correspondingpositive eigenvector, and A is an n × n nonnegative irreducible real matrix