Fixed point of a vector

Question

It seems to me that FixedPoint is designed to work with a particular value, but what if we want it to operate on a vector instead?

I start with an nxn matrix mat and the function:

f[vec_]:=Exp[-vec]/Total[Exp[-vec]]

I want to find a vector of probabilities vec={p[1], p[2], ..., p[n]} such that:

vec==f[mat.vec],

where each p[i]>0 and the sum of the p[i]'s is 1.

Is there a way to do this in general? Or let's take a specific matrix:

test={{0.5, 0.44, 0.58}, {0.56, 0.5, 0.41}, {0.42, 0.59, 0.5}}

Can I find a vector of probabilities {p1,p2,p3} that works here?

This seems like a FixedPoint type of problem, but I'd settle for any solution, like NSolve or some Module/Block. I've been puzzling over this for a while, so any help would be appreciated.

Answer 1

FixedPoint[] works with anything, including numbers, matrices, and strings of text. As long as it can compare runs. Your problem does seem to be a fixed point problem (didn't exhaustively explore initial conditions or think it through).

 mat = {{0.5, 0.44, 0.58}, 
       {0.56, 0.5, 0.41}, 
       {0.42, 0.59, 0.5}}

f[vec_] := Exp[-vec]/Total[Exp[-vec]]

FixedPoint[f[mat.#] &, {10, 100, -10}]

(*  {0.331227, 0.336697, 0.332076}  *)

Answer 2

Try this:

mat = {(1/1.52)*{0.5, 0.44, 0.58}, (1/1.47)*{0.56, 0.5, 
     0.41}, (1/1.51)*{0.42, 0.59, 0.5}};
MatrixPower[mat, 100]   (*100th power*)

Multiplying each row with a constant standardizes the rows to have a stochastic matrix.

{0.32987, 0.34012, 0.33001}