ELA, Volume 10, pp. 31-45, February 2003, abstract.
A Contribution to Collatz's Eigenvalue Inclusion
Theorem For Nonnegative Irreducible Matrices
Tedja Santanoe Oepomo
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 corresponding
positive eigenvector, and A is an nxn nonnegative
irreducible real matrix.