% operation on vectors and matrices
% matrix creation A

A = [1 2 3; 3 4 5];

% matrix transposition A' or transpose(A)
A'

% matrix multiplication 
B = [4 5; 6 7; 8 9];

A*B

% matrix elements multiplication

A.*B

% cubic power of A (trivial)

n = 3;
A^3

% A.^C
C = [ 1 2 1; 2 1 2];
A.^C

% det(A) (?!) make from the matrix A a square matrix and then find its determinant

D1 = A(:, 1:2)
det(D1)

% or

D2 = [A(:, 1), A(:, 3)]
det(D2)

% the main diagonal
diag(A)

% try to find the difference between D \ B and inv(D)*B
% when D is a rectangular matrix inv(D) does not exist 
% put 
D = [1 4; 7 2; 9 1];

% then exists
D \ B

% and

D / B
% exists as well.
 
% Let's check the interpretation of the \ or / division in the case of rectangular matrices.

% a pseudo-inverse matrix

% find size of D
s = size(D);

% find rank of D
r = rank(D);

% transpose matrix D
E = D'

% and the inverse matrix
G = inv(E*D)

% define a matrix K of [r, m] size
K = [4 5 1; 3 8 4]; 

% and finally:
Dpseudo = G*E + K*(ones(size(K, 2), size(K, 2)) - D*G*E)

% the pseudo-inverse matrix

% the matrix A:

A = [1 0 1 1; 0 -1 -1 -2; -1 1 0 1];

% the rank of A

rA = rank(A);

% as B matrix we take a matrix built from rA columns of A that are independent each other

B = A(:, 1:rA);
% check the rank of B

rB = rank(B);
% as C matrix we take
% C = [eye(2), c1, ..., c(n-rA)]
% e.g.:
% A =[1 0 -1; 0 1 0; 0 -1 0]
% B = A(:, 1:2)
% c1 => c(rA + 1)
%A(1,rA+1) = B(1, :)*c(rA+1)
%A(2,rA+1) = B(2, :)*c(rA+1)
% =>
%A(1:rA, rA+1) = B(1:rA, :)*c(rA+1) 
% => for the first vector c
%c1 = inv(B(1:rA, :))*A(1:rA, rA+1)
%and so on for others then we have C

% we are back to our example
% we need vectors c: c1 and c2

c1 = inv(B(1:rA, :))*A(1:rA, rA+1);
c2 = inv(B(1:rA, :))*A(1:rA, rA+2);

% or more generally
Bprim = inv(B(1:rA, :));

for i=1:size(A, 2)-rA
	c(:,i) = Bprim*A(1:rA, rA+i);
end

% then

C = [eye(rA), c];

% then we need to find pseudo-inverse matrices to B and C
B2 = B';
% the pseudo-inverse matrix as left inverse because rank(B) = size(B, 2)
B3 = inv(B2*B);
B4 = B3*B2;

% and for C

C2 = C';
% the pseudo-inverse matrix as right inverse because rank(C) = size(C, 1)
C3 = inv(C*C2);
C4 = C2*C3;

% and the seeked pseudo-inverse matrix of A is

A4 = C4*B4; 

% then one can check the result with the outcome of the \ operator
% let's take the matric G

G = [3 4; 1 3; 1 2];

Gout1 = A4*G;

Gout2 = A\G;

% check if they are the same:
Gout1 == Gout2

% full agreement should give matrix of (logical) ones => true