LQ decomposition ($A=LQ,Q{Q}^T=I$)

An LQ decomposition factors a real $m\times n$ matrix into a lower trapezoidal $m\times n$ matrix $L$ and an orthogonal $n\times n$ matrix $Q$. Transposing a QR decomposition of $A^T$ proves existence. Equivalently, right Householder transformations clear the part of each row beyond its diagonal while preserving earlier zeros. This factorization is useful for underdetermined systems, complementing the role of QR decomposition in overdetermined systems.