QL decomposition

Question

I've read about the $QR$-decomposition, so I wonder is there an algorithm that does the factorization ${A = QL}$ of ${A} \in \mathbb{R}^{m \times n}$, $m \geq n$, such that ${Q} \in \mathbb{R}^{m \times m}$ is orthogonal and ${L} \in \mathbb{R}^{m \times n}$ has the following form $$ {L} = \begin{pmatrix} \bar{{L}} \\ {0} \end{pmatrix} $$ where $\bar{{L}} \in \mathbb{R}^{n \times n}$ is a lower triangular matrix. If it is impossible can you explain the reasons?

Answer 1

$QR$-decomposition is a consequence of Gram-Schmidt orthogonalization. You could orthogonalize columns of $A$ starting from the last with lower triangular matrix.Then you would have to append columns to it to make it square and orthogonal. So $Q$ wouldn't be unique.