Householder QR Factorization for m by n Matrix (both m>=n and mWhy in all of books I read about numerical linear algebra (e.g. Matrix Computations by Golub and Numerical Linear Algebra and Applications by Datta and many others), Householder QR factorization have mentioned only for m by n (m rows and n columns) matrices with $m>=n$? I am looking for a trusty source that explains Householder QR factorization for a general rectangular matrix, not only for $m>=n$. I found here a MATLAB code that produce R for a general matrix, and I can get Q from it. Also MATLAB built-in qr() function produce Q and R for any rectangular matrix. But why authors of that books did not mention it?
 A: "...Householder QR factorization have mentioned only for $m$ by $n$ ($m$ rows and $n$ columns) matrices with $m\geq n$..."
That is not true, at least for Matrix Computations by Golub and Van Loan and others I know about.
It is true that the initial expositions can be a bit simplified.
The reason why these and other authors concentrate more on the case of a full-rank "tall" matrix $A$ is that problems with such matrices just appear more often in practice and are easier to handle and describe, e.g., you don't need to apply any kind of pivoting.
If this is not the case, then you must do some sort of pivoting (see, e.g., Section 5.4.1 "Rank Deficiency: QR with Column Pivoting" in the second edition of MatComp and its application in Section 5.7.2 "Underdetermined Systems"). Consider, e.g., that $A$ is "wide", that is, $m<n$ and has full row rank. Even in this case you need to do some pivoting because there's no guarantee that the leading part of $A$ has full rank (as in the other answer).
You can also find a nice treatment of general QR decompositions in the books Matrix Algorithms by Stewart (volume I) and Numerical Methods for Least Squares Problems by Bjorck.
A: Suppose $m<n$ and let $A = \begin{bmatrix} A_1 A_2\end{bmatrix}$, where $A_1$ is square. Let $A_1 = Q_1 R_1$ be a QR decomposition of $A_1$, then have
$A = Q_1 \begin{bmatrix} R_1 Q_1^TA_2\end{bmatrix}$, which is a QR decomposition of $A$.
