LQ decomposition

Question

so I'm studying for Dirty Paper Coding on MU-MIMO, the system needs LQ decomposition in it. and I want to ask, where do we get the value of L (as in lower matrix triangular) from this case? can anyone help? thank you so much

score 1 · Answer 1 · answered May 07 '19 at 21:54

Take an under-determined matrix $A$ and take the transpose, $A^{T}$. Now take the $QR$ decomposition. Now take the transpose of each $Q,R$

$ A = R^{T}Q^{T}$

Like below with some python code.

import numpy as np

A = np.random.rand(10,20)

At  = np.transpose(A)

Q,R = np.linalg.qr(At)

Rt = np.transpose(R) 
Qt = np.transpose(Q)

error = np.linalg.norm(A - np.dot(Rt,Qt))
print(error)

2.023080674308577e-15

score 0 · Answer 2 · answered Sep 03 '14 at 13:37

0

It is a generalization of QR decomposition. You can read more about LQ decomposition here.

answered Sep 03 '14 at 13:37

user153012

12,240

1

thank you, it's from gram-schmidt's methode and R matrix transpose, now I get it. thank you – Nadhia Iffah Saraswati Sep 24 '14 at 00:41
@NadhiaIffahSaraswati You're welcome. The point is not that. Here $L$ is a lower triangle matrix, and $Q$ is an orthogonal matrix. How to produce the decomposition is another question. I think analogously, like in the QR decomposition, you can use also Householder transformation or Givens rotations. – user153012 Sep 24 '14 at 00:50

LQ decomposition

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