Given a primitive stochastic matrix, we provide an upper bound on the moduli of its non-Perron eigenvalues. The bound is given in terms of the weights of the cycles in the directed graph associated with the matrix. The bound is attainable in general, and we characterize a special case of equality when the stochastic matrix has a positive row. Applications to Leslie matrices and to Google-type m...

Let $ m , n in mathbb{N}$, $D$ be a division ring, and $M_{m times n}(D)$ denote the bimodule of all $m times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We ...

We illustrate how a technique from the theory of random iterations of functions can be used within the theory of products of matrices. Using this technique we give a simple proof of a basic theorem about the asymptotic behavior of (deterministic) “backwards products” of row-stochastic matrices and present an algorithm for perfect sampling from the limiting common rowvector (interpreted as a pro...

let $ m , n in mathbb{n}$, $d$ be a division ring, and $m_{m times n}(d)$ denote the bimodule of all $m times n$ matrices with entries from $d$. first, we characterize one-sided submodules of $m_{m times n}(d)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $d$. next, we introduce the notion of a nest module of matrices with entries from $d$. we ...

A real or complex n × n matrix is generalized doubly stochastic if all of its row sums and column sums equal one. Denote by V the linear space spanned by such matrices. We study the reducibility of V under the group Γ of linear operators of the form A 7→ PAQ, where P and Q are n×n permutation matrices. Using this result, we show that every linear operator φ : V → V mapping the set of generalize...

We consider a two player finite strategic zero-sum game where each player has stochastic linear constraints. We formulate the stochastic constraints of each player as chance constraints. We show the existence of a saddle point equilibrium if the row vectors of the random matrices, defining the stochastic constraints of each player, are elliptically symmetric distributed random vectors. We furth...

The polytope Q, of the convex combinations of the permutation matrices of order n is well known (Birkhoff’s theorem) to be the polytope of doubly stochastic matrices of order n. In particular it is easy to decide whether a matrix of order n belongs to Q,. . check to see that the entries are nonnegative and that all row and columns sums equal 1. Now the permutations z of { 1, 2, . . . . n} are i...

We present a multivariate generating function for all n × n nonnegative integral matrices with all row and column sums equal to a positive integer t , the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope Bn of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for t...