404 research outputs found
Infinitely divisible nonnegative matrices, -matrices, and the embedding problem for finite state stationary Markov Chains
This paper explicitly details the relation between -matrices, nonnegative
roots of nonnegative matrices, and the embedding problem for finite-state
stationary Markov chains. The set of nonsingular nonnegative matrices with
arbitrary nonnegative roots is shown to be the closure of the set of matrices
with matrix roots in . The methods presented here employ nothing
beyond basic matrix analysis, however it answers a question regarding
-matrices posed over 30 years ago and as an application, a new
characterization of the set of all embeddable stochastic matrices is obtained
as a corollary
A Perron theorem for matrices with negative entries and applications to Coxeter groups
Handelman (J. Operator Theory, 1981) proved that if the spectral radius of a
matrix is a simple root of the characteristic polynomial and is strictly
greater than the modulus of any other root, then is conjugate to a matrix
some power of which is positive. In this article, we provide an explicit
conjugate matrix , and prove that the spectral radius of is a simple and
dominant eigenvalue of if and only if is eventually positive. For
real matrices with each row-sum equal to , this criterion can be
declined into checking that each entry of some power is strictly larger than
the average of the entries of the same column minus . We apply the
criterion to elements of irreducible infinite nonaffine Coxeter groups to
provide evidences for the dominance of the spectral radius, which is still
unknown.Comment: 14 page
The Collatz-Wielandt quotient for pairs of nonnegative operators
In this paper we consider two versions of the Collatz-Wielandt quotient for a
pair of nonnegative operators A,B that map a given pointed generating cone in
the first space into a given pointed generating cone in the second space. If
the two spaces and two cones are identical, and B is the identity operator then
one version of this quotient is the spectral radius of A. In some applications,
as commodity pricing, power control in wireless networks and quantum
information theory, one needs to deal with the Collatz-Wielandt quotient for
two nonnegative operators. In this paper we treat the two important cases: a
pair of rectangular nonnegative matrices and a pair completely positive
operators. We give a characterization of minimal optimal solutions and
polynomially computable bounds on the Collatz-Wielandt quotient.Comment: 24 pages. To appear in Applications of Mathematics, ISSN 0862-794
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
- …