clear
rng(1)

%%
% Here we let A be a 5x5 matrix. We also choose a random
% 5-vector.
A = magic(5)/65;
x = randn(5,1)

%%
% Applying matric-vector multiplication once doesn't do anything
% recognizable.
y = A*x

%%
% Repeating the multiplication still doesn't do anything obvious.
z = A*y

%%
% But if we keep repeating the matrix-vector multiplication,
% something remarkable happens: A x \approx x. 
for j = 1:8  
    x = A*x;  
end
[x, A*x]

%%
% This seems to occur regardless of the starting value of x. 
x = randn(5,1)
for j = 1:8
    x = A*x;  
end
[x, A*x]