1. ## Probability Vectors

Show that when x is a probability vector and A is a stochastic matrix, then Ax is another probability matrix

2. ## Re: Probability Vectors

You mean "then Ax is another probability vector".

What are your precise definitions of "probability vector" and "stochastic matrix"?

3. ## Re: Probability Vectors

Yes, sorry.

We say a vector X in Rn is a probability vector if all its entries are non-negative and add up to 1. We say a square matrix is a stochastic matrix if each of its column vectors is a probability vector

4. ## Re: Probability Vectors

clearly each coordinate of Ax is non-negative being the sum of products of non-negative numbers.

let us call the i,j-th entry of A, aij.

then (Ax)i = ai1x1 +...+ ainxn.

hence (Ax)1 +...(Ax)i +...+ (Ax)n =

(a11x1 +...+ a1nxn) +...+ (ai1x1 +...+ ainxn) +...+ (an1x1+ ...+ annxn) =

(a11 +...+ an1)x1 +...+ (a1j +...+ anj)xj +...+ (an1 +...+ ann)xn.

but a1j +...+ anj = 1, since A is a stochastic matrix (this is just the sum of the j-th column, which is a probability vector), for EACH j,

hence (Ax)1 +...+ (Ax)n = (a11 +...+ an1)x1 +....+ (an1 +...+ ann)xn = x1 +... + xn = 1, so Ax is a probability vector.