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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 estimating the 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] and the spectral radius of A, where x is any positive n-vector, X[A] is the correspondingpositive eigenvector, and A is an n × n nonnegative irreducible real matrix.