Let A and B be n × m matrices. The matrix B is said to be g-row majorized (respectively g-column majorized) by A, if every row (respectively column) of B, is g-majorized by the corresponding row (respectively column) of A. In this paper all kinds of g-majorization are studied on Mn,m, and the possible structure of their linear preservers will be found. Also all linear operators T : Mn,m ---> Mn...

For any stochastic matrix A of order n, denote its eigenvalues as λ1(A), . . . , λn(A), ordered so that 1 = |λ1(A)| ≥ |λ2(A)| ≥ . . . ≥ |λn(A)|. Let cT be a row vector of order n whose entries are nonnegative numbers that sum to n. Define S(c), to be the set of n × n row-stochastic matrices with column sum vector cT . In this paper the quantity λ(c) = max{|λ2(A)||A ∈ S(c)} is considered. The ve...

We provide an explicit combinatorial formula for the volume of the polytope of n× n doubly-stochastic matrices, also known as the Birkhoff polytope. We do this through the description of a generating function for all the lattice points of the closely related polytope of n × n real non-negative matrices with all row and column sums equal to an integer t. We can in fact recover similar formulas f...

Sparsified Randomization Monte Carlo (SRMC) algorithms for solving systems of linear algebraic equations introduced in our previous paper [34] are discussed here in a broader context. In particular, I present new randomized solvers for large systems of linear equations, randomized singular value (SVD) decomposition for large matrices and their use for solving inverse problems, and stochastic si...

We study the matrices Qk of in-forests of a weighted digraph Γ and their connections with the Laplacian matrix L of Γ. The (i, j) entry of Qk is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Qk, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representation...

We present a construction for tight wavelet frames for nonuniform spline type spaces. Our basic requirement is that the twoscale matrices are row stochastic as this admits the local construction of a set of framelets for each scaling function. The framelets have one vanishing moment, small support, and can be designed to have symmetry properties. We apply our method to univariate splines and to...

We study the conceptmatrix majorization: for two real matrices A and B having m rows we say that A majorizes B if there is a row-stochastic matrix X with AX = B. A special case is classical notion of vector majorization. Several properties and characterizations of matrix majorization are given. Moreover, interpretations of the concept in mathematical statistics are discussed and some combinator...

A Laplacian matrix, L = (lij) ∈ R , has nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with − 1 n ≤ lij ≤ 0 at j 6= i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrice...

Let p be a homogeneous polynomial of degree n in n variables, p(z1, . . . , zn) = p(Z), Z ∈ C. We call such a polynomial p H-Stable if p(z1, . . . , zn) 6= 0 provided the real parts Re(zi) > 0, 1 ≤ i ≤ n. This notion from Control Theory is closely related to the notion of Hyperbolicity used intensively in the PDE theory. The main theorem in this paper states that if p(x1, . . . , xn) is a homog...