% MCS 320 Project 3 Markov Chains with MATLAB
%
%    This script forms the basis for a possible solution
%  to the project.  A complete solution consists of the
%  numerical values produces by this cript and the print
%  out of the plots of the projections of the trajectories.
%
% 0. Introduction
%
%  We take a stochastic matrix A (columns sum to one)
%  and some starting point x0.
%
A = [.7 .1 .3; .2 .8 .3; .1 .1 .4]
x0 = [.8; .1; .1]
%
% 1. Assignment One
%
%  The following script computes the trajectory:
%
type trajectory.m
T = trajectory(A,x0,10)
%
% 2. Assignment Two
%
%  We generate trajectories starting at random points andj
%  make plots of the projections on the coordinate planes.
%
X0 = [rand(1,15)/2; rand(1,15)/2; ones(1,15)]
X0(3,:) = X0(3,:) - X0(1,:) - X0(2,:)
type trajplots.m
subplot(3,1,1)
trajplots(A,X0,40,1,2);  % projection on xy-plane
subplot(3,1,2)
trajplots(A,X0,40,1,3);  % projection on xz-plane
subplot(3,1,3)
trajplots(A,X0,40,2,3);  % projection on yz-plane
%
% 3. Assignment Three
%
%  Observing that all trajectories end at the same point,
%  regardless of the starting point, we compute the steady-state
%  vector via an eigenvalue problem, or more directly (since 1
%  is always an eigenvalue) via a basis for the null space.
%
[v,d] = eig(A)           % first method
v(:,1)/sum(v(:,1))       % we need to scale
nA = null(A-eye(3))      % second, more direct, method
nA/sum(nA)               % also here we scale