100 research outputs found
Cyclic classes and attraction cones in max algebra
In max algebra it is well-known that the sequence A^k, with A an irreducible
square matrix, becomes periodic at sufficiently large k. This raises a number
of questions on the periodic regime of A^k and A^k x, for a given vector x.
Also, this leads to the concept of attraction cones in max algebra, by which we
mean sets of vectors whose ultimate orbit period does not exceed a given
number. This paper shows that some of these questions can be solved by matrix
squaring (A,A^2,A^4, ...), analogously to recent findings concerning the orbit
period in max-min algebra. Hence the computational complexity of such problems
is of the order O(n^3 log n). The main idea is to apply an appropriate diagonal
similarity scaling A -> X^{-1}AX, called visualization scaling, and to study
the role of cyclic classes of the critical graph. For powers of a visualized
matrix in the periodic regime, we observe remarkable symmetry described by
circulants and their rectangular generalizations. We exploit this symmetry to
derive a concise system of equations for attraction cpne, and we present an
algorithm which computes the coefficients of the system.Comment: 38 page
Extremals of the supereigenvector cone in max algebra: a combinatorial description
We give a combinatorial description of extremal generators of the
supereigenvector cone {x: Ax>=x} in max algebra.Comment: 11 page
Multiorder, Kleene stars and cyclic projectors in the geometry of max cones
This paper summarizes results on some topics in the max-plus convex geometry,
mainly concerning the role of multiorder, Kleene stars and cyclic projectors,
and relates them to some topics in max algebra. The multiorder principle leads
to max-plus analogues of some statements in the finite-dimensional convex
geometry and is related to the set covering conditions in max algebra. Kleene
stars are fundamental for max algebra, as they accumulate the weights of
optimal paths and describe the eigenspace of a matrix. On the other hand, the
approach of tropical convexity decomposes a finitely generated semimodule into
a number of convex regions, and these regions are column spans of uniquely
defined Kleene stars. Another recent geometric result, that several semimodules
with zero intersection can be separated from each other by max-plus halfspaces,
leads to investigation of specific nonlinear operators called cyclic
projectors. These nonlinear operators can be used to find a solution to
homogeneous multi-sided systems of max-linear equations. The results are
presented in the setting of max cones, i.e., semimodules over the max-times
semiring.Comment: 26 pages, a minor revisio
On hyperplanes and semispaces in max-min convex geometry
The concept of separation by hyperplanes is fundamental for convex geometry
and its tropical (max-plus) analogue. However, analogous separation results in
max-min convex geometry are based on semispaces. This paper answers the
question which semispaces are hyperplanes and when it is possible to
classically separate by hyperplanes in max-min convex geometry
Characterizing matrices with -simple image eigenspace in max-min semiring
A matrix is said to have -simple image eigenspace if any eigenvector
belonging to the interval is the unique solution of the system in
. The main result of this paper is a combinatorial characterization of such
matrices in the linear algebra over max-min (fuzzy) semiring.
The characterized property is related to and motivated by the general
development of tropical linear algebra and interval analysis, as well as the
notions of simple image set and weak robustness (or weak stability) that have
been studied in max-min and max-plus algebras.Comment: 23 page
X-simple image eigencones of tropical matrices
We investigate max-algebraic (tropical) one-sided systems
where is an eigenvector and lies in an interval . A matrix is
said to have -simple image eigencone associated with an eigenvalue
, if any eigenvector associated with and belonging to
the interval is the unique solution of the system in
. We characterize matrices with -simple image eigencone geometrically and
combinatorially, and for some special cases, derive criteria that can be
efficiently checked in practice.Comment: 25 page
CSR expansions of matrix powers in max algebra
We study the behavior of max-algebraic powers of a reducible nonnegative n by
n matrix A. We show that for t>3n^2, the powers A^t can be expanded in
max-algebraic powers of the form CS^tR, where C and R are extracted from
columns and rows of certain Kleene stars and S is diadonally similar to a
Boolean matrix. We study the properties of individual terms and show that all
terms, for a given t>3n^2, can be found in O(n^4 log n) operations. We show
that the powers have a well-defined ultimate behavior, where certain terms are
totally or partially suppressed, thus leading to ultimate CS^tR terms and the
corresponding ultimate expansion. We apply this expansion to the question
whether {A^ty, t>0} is ultimately linear periodic for each starting vector y,
showing that this question can be also answered in O(n^4 log n) time. We give
examples illustrating our main results.Comment: 25 pages, minor corrections, added 3 illustration
On the max-algebraic core of a nonnegative matrix
The max-algebraic core of a nonnegative matrix is the intersection of column
spans of all max-algebraic matrix powers. Here we investigate the action of a
matrix on its core. Being closely related to ultimate periodicity of matrix
powers, this study leads us to new modifications and geometric
characterizations of robust, orbit periodic and weakly stable matrices.Comment: 27 page
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