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In a recent paper the authors proposed a lower bound on , where , , is an eigenvalue of a transition matrix of an ergodic Markov chain. The bound, which involved the group inverse of , was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when...
This paper is concerned with classification criteria, asymptotic behaviour and stationarity of a non-Markovian model with linear transition rule, called a linear OM-chain. This problems are solved by making use of the structure of the stochastic matrix appearing in the definition of such a model. The model studied includes as special cases the Markovian model as well as the linear learning model, and has applications in psychological and biological research, in control theory, and in adaptation...
Let A be an n×n irreducible nonnegative (elementwise) matrix. Borobia and Moro raised the following question: Suppose that every diagonal of A contains a positive entry. Is A similar to a positive matrix? We give an affirmative answer in the case n = 4.
In this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko–Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.
Consider a non-centered matrix with a separable variance profile:
Matrices and are non-negative deterministic diagonal, while matrix is deterministic, and is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by the resolvent associated to , i.e.
Given two sequences of deterministic vectors and with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:
as the dimensions...
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