favouriteMatrix = [1,2,3;4,5,6;7,8,9] 
% define a matrix

det(favouriteMatrix) 
% determinant of that matrix

rand(3) 
% creates a 3x3 matrix of randomly distributed numbers on (0,1)

help rand % open help file for rand
help randi % open help file for randi

James = randi([1,10],50,50);
% we now have a 50x50 matrix called James, who contains random integers
% between 1 and 10

E = eig(James)
% eigenvalues of James (outputs a list, which we have labelled E)

figure
% create figure

plot(real(E),imag(E),'bo')
% plot eigenvalues of James on the complex plane, using stylish blue
% circles
% i.e. plot real component against imaginary component of each item in E

p=0.3; q=1-p; 
% define transition rates

A = [0,p,q;q,0,p;p,q,0];
% create 3x3 stochastic transition matrix (with periodic boundaries)

eig(A)
% eigenvalues!

figure
% create new figure

plot(real(eig(A)),imag(eig(A)),'bo')
% plot those eigenvalues

hold on % hold the plot!

ang = [0:0.01:2*pi];
%create list of values between 0 and 2*pi in increments of 0.01

plot(0+1.*cos(ang),0+1.*sin(ang),'r')
% parametric plot of a circle radius 1, centered on (0,0)
% plotted as a fashionable red line

A

A^2

A^3

A^4

A^5

A^10

A^100

% iterate the matrix and watch what happens to it's entries
% what does this tell you?

[V,E] = eig(A')
% obtain the eigenvectors and eigenvalues of A
% N.B. the transpose is necessary to obtain the LEFT eigenvectors
% we aren't interested in RIGHT eigenvectors

r = 0.1; p=0.2;q=1-p-r; 
% define new transition rates

A = [r,p,0,q;q,r,p,0;0,q,r,p;p,0,q,r]
% create 'lazy' random walk transition matrix 

A^2

A^10

A^100

% iterate the matrix...

figure
%create new figure

plot(real(eig(A)),imag(eig(A)),'bo') % plot eigenvalues of A (blue circles)
hold on; 
plot(0.1+0.9.*cos(ang),0.9.*sin(ang),'r') % plot radius 0.9 circle centered on (0.1,0)
plot(real(eig(A^10)),imag(eig(A^10)),'go') % plot eigenvalues of A^10 (green circles)
% visualisation of Gershgorin Theorem