We will use Householder reflections to produce a QR factorization of the matrix
A = magic(6);
A = A(:,1:4);
[m,n] = size(A);
Our first step is to introduce zeros below the diagonal in column 1. Define the vector
z = A(:,1);
Applying the Householder definitions gives us
v = z - norm(z)*eye(m,1);
P = eye(m) - 2/(v'*v)*(v*v'); % reflector
By design we can use the reflector to get the zero structure we seek:
P*z
Now we let
A = P*A
We are set to put zeros into column 2. We must not use row 1 in any way, lest it destroy the zeros we just introduced. To put it another way, we can repeat the process we just did on the smaller submatrix
A(2:m,2:n)
z = A(2:m,2);
v = z - norm(z)*eye(m-1,1);
P = eye(m-1) - 2/(v'*v)*(v*v'); % reflector
We now apply the reflector to the submatrix.
A(2:m,2:n) = P*A(2:m,2:n)
We need two more iterations of this process.
for j = 3:n
z = A(j:m,j);
k = m-j+1;
v = z - norm(z)*eye(k,1);
P = eye(k) - 2/(v'*v)*(v*v');
A(j:m,j:n) = P*A(j:m,j:n);
end
We have now reduced the original 
to an upper triangular matrix using four orthogonal Householder reflections:
to an upper triangular matrix using four orthogonal Householder reflections:R = A