MATLAB gives direct access to both the full and thin forms of the QR factorization.
A = magic(5);
A = A(:,1:4);
[m,n] = size(A)
m =
5
n =
4
Here is the full form:
[Q,R] = qr(A);
szQ = size(Q)
szQ =
5 5
szR = size(R)
szR =
5 4
We can test that is orthogonal.
QTQ = Q'*Q
QTQ =
1.0000e+00 5.2681e-17 7.5106e-17 2.6976e-17 -2.3095e-17 5.2681e-17 1.0000e+00 3.9971e-17 -3.3699e-17 8.5462e-18 7.5106e-17 3.9971e-17 1.0000e+00 -1.5267e-17 -1.8197e-16 2.6976e-17 -3.3699e-17 -1.5267e-17 1.0000e+00 -2.0695e-17 -2.3095e-17 8.5462e-18 -1.8197e-16 -2.0695e-17 1.0000e+00
norm(QTQ - eye(m))
ans =
3.7889e-16
With a second input argument given, the thin form is returned.
[Q,R] = qr(A,0);
szQ = size(Q)
szQ =
5 4
szR = size(R)
szR =
4 4
Now cannot be an orthogonal matrix, because it is not even square, but it is still ONC.
Q'*Q - eye(4)
ans =
-2.2204e-16 5.2681e-17 7.5106e-17 2.6976e-17 5.2681e-17 2.2204e-16 3.9971e-17 -3.3699e-17 7.5106e-17 3.9971e-17 -2.2204e-16 -1.5267e-17 2.6976e-17 -3.3699e-17 -1.5267e-17 -2.2204e-16