An LQ decomposition of a real 
matrix 
with 
is a matrix decomposition
 |
where 
is an 
lower triangular matrix and the 
matrix 
has rows forming an orthonormal
set, so that 
.
This is the reduced form. In the full form, the factor 
is an 
orthogonal matrix
and the left factor is 
. For complex matrices,
conjugate transpose replaces transpose,
and the full factor is a unitary matrix.
An LQ decomposition follows by transposing a QR decomposition of 
.
If 
and 
has matrix rank 
, it gives the solution of 
having the smallest Euclidean
norm,
 |
using the reduced factors. In computation, the linear system of equations with matrix 
is solved without forming its matrix
inverse.
