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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 019 243 237
Author(s):
Oepomo, Tedja Santanoe:
Shorttitle:
A contribution to Collatz' eigenvalue inclusion theorem for non-negative irreducible matrices
Source:
Electron. J. Linear Algebra 10, 31-45, electronic only (2003): http://www.emis.de/journals/ELA/ela-articles/10.html
Classification:
 15A48 Positive matrices and their generalizations; cones of matrices
Primary Classification:
 15A18 Eigenvalues, singular values, and eigenvectors
Secondary Classification:
 15A42 Inequalities involving eigenvalues and eigenvectors 65F15 Eigenvalues, eigenvectors
Keywords:
real matrices; irreducible; non-negative; spectral radius; positive eigenvector; Collatz-Wielandt estimates; equiconvergence property; wege shaped domain of inclusion intervals
Review:

Many n by n matrices A of practical interest are irreducible
(i.e., not reducible to a block-triangular form by a permutation
of basis) and non-negative (i.e., they are composed of
non-negative elements). Many people paid attention to their
spectral radius \Lambda[A] and to the related (``ground-state" or
``Perron") positive eigenvector X[A]. The author pays his/her
main attention to the ``coherence" (i.e., simultaneous closeness)
of the Collatz-Wieland lower and upper estimates m(x) and M(x) of
\Lambda[A] (forming and ``inclusion interval") for variable
positive x's. Working with the ``max" norm, there estimates are
naturally defined as the respective maximum and minimum of the
components of the vector f(x) of quotients of individual
components of Ax and x, and their relevant properties are
summarized. Several apparently new results (especially, the
uniqueness of the maximizer x in Theorem 2.2 and its
equiconvergence with m(x) [or, mutatis mutandis, with M(x)] in
Theorem 2.4) are also added and proved, implying that the set of
all the inclusion intervals forms a two-dimensional wedge-shaped
domain (this is the main result - see Figure 1 and Theorem 3.2).
Remarks to the editors:
Due to temporary change of e-address, editor and PC, I used non-tex \Lambda etc (with apologies), and a less complete set of classification numbers than usual.

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