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.