LQ decomposition (HP compatible)

15.3.4 LQ decomposition (HP compatible)

The LQ decomposition of a matrix A is A=LQP, where L is lower triangular the same size as A (if A is not square, then ℓ_i,j=0 for i>j), Q is an orthogonal matrix, and P is a permutation matrix.

The LQ command finds the LQ decomposition of a matrix.

LQ takes A, a matrix.
LQ(A) returns a list [L,Q,P] of the matrices given by the LQ decomposition.

Examples

L,Q,P:=LQ([[4,0,0],[8,-4,3]])

⎡
⎢
⎢
⎣

⎡
⎢
⎣

4.0	0.0	0.0
8.0	5.0	−4.4408920985×10⁻¹⁶

⎤
⎥
⎦

⎡
⎢
⎢
⎣

1.0	0.0	0.0
0.0	−0.8	0.6
0.0	−0.6	−0.8

⎤
⎥
⎥
⎦

⎡
⎢
⎢
⎣

1	0	0
0	1	0
0	0	1

⎤
⎥
⎥
⎦

L,Q,P:=LQ([[24,18],[30,24]])

⎡
⎢
⎣

−30.0	0.0
−38.4	−1.2

⎤
⎥
⎦

⎡
⎢
⎣

−0.8	−0.6
0.6	−0.8

⎤
⎥
⎦

⎡
⎢
⎣

1	0
0	1

⎤
⎥
⎦

In the above examples, LQ=PA.