We determine analytically the condition number of the PageRank problem. Specifically, we prove the following statement: “Let P be an n × n row-stochastic matrix whose diagonal elements Pii = 0. Let c be a real number such that 0 ≤ c < 1. Let E be the n × n rank-one row-stochastic matrix E = ev , where e is the n-vector whose elements are all ei = 1, and v is an n-vector that represents a probab...

We consider the web hyperlink matrix used by Google for computing the PageRank whose form is given by A(c) = [cP + (1 − c)E]T , where P is a row stochastic matrix, E is a row stochastic rank one matrix, and c ∈ [0, 1]. We determine the analytic expression of the Jordan form of A(c) and, in particular, a rational formula for the PageRank in terms of c. The use of extrapolation procedures is very...

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...

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...

The transition matrix, which characterizes a discrete time homogeneous Markov chain, is a stochastic matrix. A stochastic matrix is a special nonnegative matrix with each row summing up to 1. In this paper, we focus on the computation of the stationary distribution of a transition matrix from the viewpoint of the Perron vector of a nonnegative matrix, based on which an algorithm for the station...

For vectors $X, Yin mathbb{R}^{n}$, we say $X$ is left matrix majorized by $Y$ and write $X prec_{ell} Y$ if for some row stochastic matrix $R, ~X=RY.$ Also, we write $Xsim_{ell}Y,$ when $Xprec_{ell}Yprec_{ell}X.$ A linear operator $Tcolon mathbb{R}^{p}to mathbb{R}^{n}$ is said to be a linear preserver of a given relation $prec$ if $Xprec Y$ on $mathbb{R}^{p}$ implies that $TXprec TY$ on $mathb...

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...