% Brief 2-by-2 Matrix Tutorial

% NEUR 1680 Spring 2008

%%  random 2 by 2 matrix

clear

W = rand(2,2)
% Note: entries are between 0 and 1


%% matrices are generalized numbers:
%   they can be worked ON: addition, multiplication, inverse...

W_inverse = inv(W)

eye(2)/W
W^-1
W.^-1

W*W_inverse
W_inverse*W

det(W)
% determinant has to be nonzero for matrix to be invertible
%   is non-zero with probability 1 for random matrix

% create matrices with 0 determinant...
%   characterize such matrices...


%% matrix operates on random vector

% Matrices can also be used as OPERATORS acting on vectors

% As such, a matrix is a LINEAR transformation of the plane


u = rand(2,1) - [.5;.5];
% Note: entries are between -.5 and .5

v = W*u;

figure(1)
plot(u(1),u(2),'b*',v(1),v(2),'r*','markersize',10)
axis([-1 1 -1 1])
axis square
grid
legend('original', 'transformed')

%% matrix operates on cursor-defined vectors

%   choose your input vectors in "interesting" ways: lines, squares, circles,...

%   click in gray area to terminate

close(1)
u = [0;0];
figure(1)
axis([-1 1 -1 1])
axis square
grid
hold on

while abs(u) <= 1

    u = ginput(1)';
    v = W*u;
    plot(u(1),u(2),'b*',v(1),v(2),'r*','markersize',10)
    legend('original', 'transformed')

end

hold off

%  based on your observations, characterize the matrix as a geometric transformation... 


%% reflection about x-axis

W = [1  0
     0 -1];
 
%% reflection about y-axis

W = [
        ];
 
%% clockwise rotation about the origin

theta = pi/3;

W = [cos(theta)  sin(theta)
     -sin(theta) cos(theta)];
 
 
% compute inverse and determinant...

%% reflection about first diagonal

W = [
        ];

%% how would you characterize the following transformation?    
 
W = [1  0
     2  1];

%% eigenstuff

[V,D] = eig(W);         % eigenvalues and eigenvectors of matrix
disp('eigenvalues:')
disp(num2str(D(logical(eye(2)))'))
disp('eigenvectors:')
disp(num2str(V))


%%  random SYMMETRIC 2 by 2 matrix

clear

W = rand(2,2);
W = .5*(W + W');

% Note: entries are still between 0 and 1

% what can you say of its eigenvalues and eigenvectors?

% why are symmetric matrices important in models of Hebbian plasticity?