stochastic matrices

an nxn matrix A is stochastic if

1. its entries are all greater than or equal to 1

2. each column’s entries sum to 1

We say a matrix A is positive stochastic if

1. A is stochastic

2. all entries of a are greater than 0

positive stochastic matrices are also called

change of state matrices

to study long term behavior, we need

high powers of A → vn = A^nv0

lambda = 1

is an eigenvalue for every stochastic matrix

If A is positive stochastic,

all other eigenvalues (besides 1) are less than 1 in magnitude

A^nv0 is guaranteed

to approach the 1-eigenspace as n approaches infinity

Steady state vector for stochastic matrix A

eigenvector for lambda = 1 whose entries are positive and sum to 1

Perron-Frobenius Theorem

If A is positive stochastic, then it has a unique steady-state vector which spans the 1-eigenspace

the 1-eigenspace for positive stochastic matrices

is a line

If v0 is any vector in R^n, then A^nv0 equals

the sum of entries of v0 times the steady state vector

we can tell the long term behavior or A^nv0

just by knowing the sum of the entries of v0

if the entries of the basis of the 1-eigenspace do not sum to 1, determine the steady state vector by

multiplying (1/sum of entries) by the basis vector

True or False: If A is a stochastic matrix, then 1 must be an eigenvalue of A

True. For a matrix to be stochastic, it must have 1 as an eigenvalue

True or False: If A is a square matrix, then A and the transpose of A must have the same eigenvalues

True.

True or False: If A is a stochastic matrix, then its 1-eigenspace must be a line

False. This is only true for positive stochastic matrices

True or False: If A is a positive stochastic matrix, then repeated multiplication by A pushes each vector toward the 1-eigenspace

True.