>> eigshow             % run this program to visualize eigenvalues/eigenvectors

>> A = [1 1; 0 2]      % demo of the "eig" command in MATLAB
A =
     1     1
     0     2
>> lambda = eig(A)     % with one argument, eig returns only the eigenvalues
lambda =
     1
     2
>> [V,D] = eig(A)      % with two arguments, eig returns matrices V and D
                       % with eigenvectors as the columns of V and 
                       % eigenvalues on the diagonal entries of D.
V =
    1.0000    0.7071
         0    0.7071
D =
     1     0
     0     2

>> [A*V V*D]           % note that A*V = V*D
ans =
    1.0000    1.4142    1.0000    1.4142
         0    1.4142         0    1.4142
>> A*V-V*D
ans =
     0     0
     0     0

>> v1 = V(:,1)         % first column of V
v1 =
     1
     0
>> v2 = V(:,2)         % second column of V
v2 =
    0.7071
    0.7071
>> 
>> lam1 = D(1,1)       % first eigenvalue
lam1 =
     1
>> lam2 = D(2,2)       % second eigenvalue
lam2 =
     2

>> [A*v1 lam1*v1]    
ans =
     1     1
     0     0
>> A*v1 - lam1*v1      % indeed, lam1 and v1 are an eigenvalue/eigenvector pair
ans =
     0
     0

>> [A*v2 lam2*v2]
ans =
    1.4142    1.4142
    1.4142    1.4142
>> A*v2 - lam2*v2      % indeed, lam2 and v2 are an eigenvalue/eigenvector pair
ans =
     0
     0


>> A = [1 1; -1 1]     % new matrix  
A =
     1     1
    -1     1

>> [V,D] = eig(A)      % compute eigenvalues/eigenvectors
V =
   0.7071             0.7071          
        0 + 0.7071i        0 - 0.7071i
D =
   1.0000 + 1.0000i        0          
        0             1.0000 - 1.0000i

% A has real entries....
% but complex eigenvalues and eigenvectors

>> v1 = V(:,1); v2 = V(:,2);
>> lam1 = D(1,1); lam2 = D(2,2);

>> [A*v1 lam1*v1]
ans =
   0.7071 + 0.7071i   0.7071 + 0.7071i
  -0.7071 + 0.7071i  -0.7071 + 0.7071i
>> A*v1 - lam1*v1
ans =
     0
     0

>> [A*v2 lam2*v2]
ans =
   0.7071 - 0.7071i   0.7071 - 0.7071i
  -0.7071 - 0.7071i  -0.7071 - 0.7071i
>> A*v2 - lam2*v2
ans =
     0
     0
 
% In fact, "typical" matrices with real entries have complex eigenvalues.
% Next we look at an example of a random matrix.

>> A = randn(100)/sqrt(100);   % entries normally distributed, mean 0, variance 1/100
>> lam = eig(A);
>> plot(real(lam),imag(lam),'k.')
>> axis equal, axis([-1.25 1.25 -1.25 1.25])

% As the matrix dimension gets large, these eigenvalues tend to cover
% the unit circle in the complex plane with uniform probability.
% To see this, we plot the unit circle...

>> T = exp(linspace(0,2i*pi,500));
>> hold on, plot(real(T),imag(T),'r-','linewidth',2)

% repeat this experiment for a larger matrix
>> A = randn(500)/sqrt(500);
>> lam = eig(A);
>> figure
>> plot(real(lam),imag(lam),'k.')
>> axis equal, axis([-1.25 1.25 -1.25 1.25])
>> hold on, plot(real(T),imag(T),'r-','linewidth',2)

% Random SYMMETRIC matrices act very differently.  Here's one example.

>> Asym = A + A';      
>> norm(Asym-Asym')    % If Asym = Asym', then this norm should be zero
ans =
     0
>> lam = eig(Asym);
>> figure
>> plot(real(lam),imag(lam),'k.')
% Now all the eigenvalues look to be real numbers!  Let's check this:
>> max(abs(imag(lam)))
ans =
     0

% This turns is a remarkable property:
% real symmetric matrices ALWAYS have real eigenvalues.