6.49.4 LQ decomposition (HP compatible): LQ
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
ℓi,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 one argument:
A, a matrix.
- LQ(A) returns a list [L,Q,P] of the matrices
given by the LQ decomposition.
Examples.
-
Input:
L, Q, P:= LQ([[4,0,0],[8,-4,3]])
Output:
⎡
⎢
⎢
⎣ | ⎛
⎜
⎝ | 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 |
| ⎞
⎟
⎟
⎠ |
| , | ⎡
⎢
⎢
⎣ | | ⎤
⎥
⎥
⎦ | ⎤
⎥
⎥
⎦ |
Here, L*Q is the same as P*A.
- Input:
L,Q,P:=LQ([[24,18],[30,24]])
Output:
Again, L*Q = P*A.