From: Kurt Foster <kfoster@rmi.net>
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!
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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  <kfoster@rmi.net> 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).
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