# Questions tagged [stochastic-matrices]

A right stochastic matrix is a nonnegative square matrix with each row summing to 1. A left stochastic matrix is a nonnegative square matrix with each column summing to 1. A doubly stochastic matrix is both right stochastic and left stochastic.

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### Power iteration error for eigenvectors

I'm trying to use the ideas of power iteration to approximate stationary distributions of Markov chains. Let's say I have a left irreducible stochastic matrix $A$ (i.e., non-negative and each column ...
As in the title, I'm searching for the spectrum of a matrix which has only one $1$ in each row (and zeros as other entries) and also that is not necessary a permutation matrix. For example, $$A =\... 1answer 24 views ### Convexity of the product of two convex sets of non-square stochastic matrices Suppose X\subseteq\mathbb{R}^{l\times m} and Y\subseteq\mathbb{R}^{m\times n} are sets of matrices with real entries. Moreover, X is the set of all l\times m stochastic matrices (each column ... 0answers 54 views ### Inverse of a stochastic matrix: when is there a strictly negative element? Let the square matrix A be invertible and left-stochastic (each column adds up to one). Under which conditions is at least one element of the inverse matrix A^{-1} strictly negative? For ... 0answers 49 views ### Spectral radius of the product of a block-diagonal matrix and a row stochastic matrix Let W\in\mathbb{R}^{N\times N} be a right (row) stochastic matrix with non-negative ij entries w_{ij}\geq0, where \sum_{j=1}^N w_{ij} = 1, and let A\in\mathbb{R}^{nN\times nN} be a block-... 2answers 821 views ### How can a Markov chain have more than one but a finite amount of stationary distributions? Here's my understanding of it: Assume we have an n\times n stochastic matrix P that represents our Markov chain such that x and y are stationary distributions for P. Then P(x) = x P(y) ... 2answers 66 views ### Formula for powers of 2\times 2 matrices Is there a formula for expanding powers of 2\times 2 matrices? I know that can be done by diagonalizing and then using the spectral decomposition of the matrix, but is there a general formula for A^... 0answers 83 views ### Upper bound on the magnitude of eigenvalue of a stochastic matrix What is the upper bound for the magnitude of the second largest eigenvalue of the following square matrix X of size d\times d? The entries are given as: X[0,0] = \alpha \in (0,1/d], X[0, d-... 1answer 83 views ### Eigenvector of stochastic matrix P is stochastic matrix, v is stochastic vector, \frac{1}{n}(v + vP + vP^{2} + \cdots + vP^{n}) \to u. Prove that uP = u. I understood that I need to show that u is eigenvector with eigenvalue ... 1answer 41 views ### Steady state probabilities of divergent Markov matrices I have a problem for which the Markov matrix turns out to be the following:$$P = \begin{pmatrix} 0 & 0.5 & 0 & 0.5\\ 0.5 & 0 & 0.5 & 0 ...
I'm trying to find an explicit formula of all the eigenvalues for the following $n$ by $n$ stochastic matrix (sum of each row/column is one): \begin{bmatrix}0&\frac{1}{n-1}&\frac{1}{n-1}&...