随机矩阵

Introduction

Difference Equations

Stochastic Matrices and the Steady State

Google’s PageRank Algorithm

Which of the following matrices is stochastic?

Let $A$ be a stochastic matrix (columns sum to $1$). Which statement is true?

In one word: a matrix with nonnegative entries whose columns sum to $1$ is called a ______ matrix.

Start with $u_1 = \begin{pmatrix}0.8\\0.2\end{pmatrix}$ and multiply by the stochastic matrix $A = \begin{pmatrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{pmatrix}$. Enter $u_2 = Au_1$ as two decimals separated by a comma (for example: 0.5,0.5).

Let

The columns sum to $1$, and $A$ has eigenvalues $0$, $\tfrac{1}{5}$ and $1$. The eigenvector of $A$ for $\lambda = 1$ is proportional to $(3,3,4)$. Starting from $u_0 = (1,0,0)$, the iterates $A^k u_0$ converge to the steady state. Enter it as three decimals separated by commas (for example: 0.5,0.5,0.5).

Let $A$ be a stochastic matrix, with nonnegative entries and columns summing to $1$. Which eigenvalue does $A$ always have?

Let

The steady state vector $w$ has $Aw = w$. Enter the ratio $w_1 / w_2$ as a fraction like $a/b$.