QR decomposition, completing Q
Potentially stupid question but bear with me...
So, after QR factorization [after Gram-Schmidt and defining R], we got an mxn-matrix A that is equal an mxn-matrix Q times an nxn-matrix R
A = QR
Now, script states "we may now complete the n orthonormal Vectors q1,...,qn [which, by the way, are the columns of Q] to an orthonormal basis of E^m [E being either the real or the complex numbers]"
So ... how do I do that?
Do I simply take any linearly independent m-vectors and add it to Q or need I calculate the missing m-n vectors and if the latter, how?
Note: It might be conducive to know that after completing Q to an mxm-matrix Q' we add m-n 0-row-vectors to R (becoming an mxn-matrix R') so as to get what the author of the scipt calls the actual QR decomposition:
A = Q'R'
Any help appreciated