Hi Douji,
It looks like you're facing some issues with the correctness of R and the absence of Q while implementing the Gram-Schmidt process.
Briefly the QR factorization can be broken down into following steps:
- Orthogonalization of columns: For each column of matrix A, subtract the projection of
onto all previously computed 
using:
v = v - (Q(:, j)' * A(:, i)) * Q(:, j);
2. Normalization for Orthonormal Vectors: After making v orthogonal to all prior vectors, normalize v to get the next column 
of Q:
of Q:Q(:, i) = v / norm(v);
3. Constructing R: Once all columns of Q are computed, R can be computed at once as (follows from the fact that 
:
:R = Q' * A;
Combining the above steps QR factorization can be computed as:
function [Q, R] = qr_factorization(A)
[m, n] = size(A);
Q = zeros(m, n);
for i = 1:n
% Compute the ith column of Q
v = A(:, i);
for j = 1:i-1
% Subtract the projection of A(:,i) onto the previous Q vectors
v = v - (Q(:, j)' * A(:, i)) * Q(:, j);
end
% Normalize to get q_i
Q(:, i) = v / norm(v);
end
% Compute R as Q' * A
R = Q' * A;
end
% Example Matrix
A = [1, 1, 1; 1, 2, 3; 1, 3, 6];
[Q, R] = qr_factorization(A);
disp('Computed Q:');
disp(Q);
disp('Computed R:');
disp(R);
