%----------------------------------------------------------%
% @Description: Orthogonalize a given set of vectors 
%               via Gram-Schmidt process
% @Input:       X   -   matrix/set of vectors x_i
% @Output:      Y   -   matrix/set of orthogonalized 
%                       vectors y_i
%----------------------------------------------------------%
function Y = gramSchmidt(X) 

%get number of Vectors
n = size(X,2);

%allocate memory
Y = zeros(size(X,1),n);

%set inital vector
Y(:,1) = X(:,1);

for k = 2:n
    %calculate sum
    sum = 0;
    for i = 1:(k-1)
        %get y_i
        yi = Y(:,i);
        %add i-th term
        sum = sum + yi.'*X(:,k) / (yi.'*yi) * yi; 
    end    
    %set the value of y_k
    Y(:,k) = X(:,k) - sum;
end