% matrix_alg_demo.m      by Andy Ruina 10/2/96
% This is a demo of some of MATLABS abilities with
% matrices: Multiplying, solving equations, creating
%           identity, finding inverse, finding eigen-values 
%           and vectors, etc.

% What is here should mostly get you by in 293 for
% matrix skills. Understanding what these commands do 
% (not necessarily how they do it) is most of what you need to know
% about matrices. 

% If you execute this file it will create a bunch of output in the
% command window. Look at it to help you understand the commands.
% See  Pratap pgs 41-58 and 64-67.
% Try online help with any of the command names you
%     see below  (eg. type  >help eye), 
%     or >help punct for help with , ; ' etc.
% 

A = [ 1 2;           % CREATING A MATRIX 
     pi 4 ]

B = [4 5;  6 7]      % CREATE ANOTHER MATRIX

C = A'               % TRANSPOSE A MATRIX
	 
b = [ 1 3 ]'         % CREATE A COLUMN VECTOR

C = A * B            % MULTIPLY ONE MATRIX BY ANOTHER  
                     % (by the rules of matrix multiplication,
					 %  ie, c(1,2) = a(1,1)*b(1,2) + a(1,2)*b(2,2)
					 
y = A * b            % MULTIPLY A MATRIX AND A COLUMN VECTOR
					 
D = A .* B           %  MULTIPLY CORRESPONDING COMPONENTS
                     %  (ie,   d(1,2) = a(1,2) * b(1,2)  )
					
comp = D(2,1)        % GET A COMPONENT OF A MATRIX
col  = D(:,2)        % GET A COLUMN OF A MATRIX (2nd col in this case)
row  = D(1,:)        % GET A ROW  (1st row in this case)

gee  = diag(D)       % GET DIAGONAL, make a column vector out of
                     % the diagonal elements of a square matrix

whiz = diag(gee)     % MAKE DIAGONAL, make a diagonal matrix (all
                     % zeros off the diagonal) out of a column vector
					 
identity = eye(4)    % MAKE AN IDENTITY MATRIX
                     % (one on the diagonals, zero elsewhere).
					 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Here are some really powerfull cool commands...
%                     ... the commands that put the MAT in MATLAB.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (By the way, all of these work fine even if compex numbers are involved.)

Adet = det(A)        % DETERMINANT calculation, less useful than some claim
                    
Ainv = inv (A)            % Finds the INVERSE of A (if it exists)
Ainv = A^(-1)             % Also finds the INVERSE of A.
A_times_inverse = A*Ainv  % CHECK. Should give identity matrix.

x = Ainv*b           % SOLVES  A*x=b for a given A and b (find Ainv first)
x = A\b              % SOLVES  A*x=b for a given A and b  more efficiently
                     % It the columns of A are not 'independent' the
					 % command tells you that your matrix is 'singular'	
					 % and doesn't give you the solution (even if there 
					 % is one).
A*x - b              % CHECKs the answer above.  Since A*x is b this 
                     % gives zero.

[VECT, VAL] = eig(A) % EIGENVECTORS and EIGENVALUES.  VECT is a matrix
                     % where each column is an eigenvector of A. VAL
					 % is a diagonal matrix where each element is the
					 % corresponding eigenvalue.  
					 % (An 'eigenvector' of a matrix A is a vector v so that
					 % A*v is proportional to v.  The proportionality 
					 % constant is the 'eigenvalue' for that eigenvector.)
					 
values = eig(A)      % EIGENVALUES of a matrix listed in a column vector
                     % So, for example, VAL (above) is the same as diag(eig(A)).

A * VECT(:,1) - VAL(1,1)*VECT(:,1)  % This checks the first eigenvector and value.
                     % If you have followed up to here you should be able
					 % to follow why this command gives two zeros (or very
					 % nearly so --- note the 1e-15).
A*VECT - VECT*VAL    % Check all  eigenvectors and values at once. (Should
                     % get very close to 4 zeros for a 2x2 matrix.) 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% If this all makes sense to you than you are doing great.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%