From: Kurt Foster Subject: Q re spectral radius Date: 5 Feb 1999 17:42:03 GMT Newsgroups: sci.math.num-analysis Keywords: Non-negative matrices I'm dealing with square matrices M having non-negative real entries, with zeroes along the diagonal. When M is symmetric, all is sweetness and light. But when it's not, hoo-boy! I can get bounds from the Gershgorin Circle Theorem and Schur's inequality. I've got some results due to Frobenius and Collatz concerning positive real eigenvalues, and vectors with positive components. If I knew that the spectral radius was the absolute value of a *real* eigenvalue, I'd be happy. Are there any simple sufficient conditions for this, when M has non-negative real entries but is not symmetric? If the spectral radius is not the absolute value of any real eigenvalue of M, how may it be estimated? And what (if anything) intelligent or useful can be said about the behavior of the sequence of magnitudes of real vectors of the type v, Mv, MMv, MMMv, etc. regarding the spectral radius? This is all so nice when M is symmetric! ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Q re spectral radius Date: 5 Feb 1999 17:20:55 -0500 Newsgroups: sci.math.num-analysis In article <79fahb\$cs4\$1@news1.rmi.net>, Kurt Foster wrote: > I'm dealing with square matrices M having non-negative real entries, >with zeroes along the diagonal. When M is symmetric, all is sweetness and >light. But when it's not, hoo-boy! I can get bounds from the Gershgorin >Circle Theorem and Schur's inequality. I've got some results due to >Frobenius and Collatz concerning positive real eigenvalues, and vectors >with positive components. [...] I am not sure how much of Frobenius's results you found, but there is a basis of knowledge in the book Richard S. Varga: Matrix Iterative Analysis, Prentice-Hall 1962 (my copy is too old to have an ISBN). Some keywords for the study of non-negative matrices and their spectral properties are directed graph of a matrix, irreducible and reducible matrices, reduction to block upper triangular form, cyclic and primitive irreducible matrices, diagonal dominance, M-matrices For a definitive refinement of Gershgorin disk theorems, find articles written by R.S. Varga and Bernhard Levinger (keywords: Minimal Gershgorin Disks). Pacif. J. Math. 15(1965), 719-729 and 17(1966), 199-210. For infinite dimensional extensions, look up my articles "Spectrum Localization in Banach Spaces" I and II, Linear Algebra and its Applications, 8, 225-236(1974) and 12, 223-229(1975), and find more from the bibliography therein. Good luck, ZVK(Slavek). ==============================================================================