I'm (reasonably) familiar with factoring a positive definite matrix $\mathbf{P} = \mathbf{L} \mathbf{L}^T = \mathbf{R}^T \mathbf{R}$, and is supported by MATLAB and Eigen.
However, I have also seen a factorization of the (same) $\mathbf{P} = \mathbf{U} \mathbf{U}^T = \mathbf{L'}^T \mathbf{L'}$
The following illustrates:
>> A = rand(3, 4)
A =
0.2785 0.9649 0.9572 0.1419
0.5469 0.1576 0.4854 0.4218
0.9575 0.9706 0.8003 0.9157
>> P = A * A.'
P =
1.9449 0.8288 2.0991
0.8288 0.7374 1.4513
2.0991 1.4513 3.3379
>> R = chol(P)
R =
1.3946 0.5943 1.5052
0 0.6198 0.8982
0 0 0.5153
% This function computes such that U * U.' = A * A.'
% Part of: http://www.iau.dtu.dk/research/control/kalmtool2.html
>> U = triag(A)
U =
-0.7475 0.2571 -1.1489
0 -0.3262 -0.7944
0 0 -1.8270
>> P2 = R.' * R
P2 =
1.9449 0.8288 2.0991
0.8288 0.7374 1.4513
2.0991 1.4513 3.3379
>> P3 = U * U.'
P3 =
1.9449 0.8288 2.0991
0.8288 0.7374 1.4513
2.0991 1.4513 3.3379
I haven't seen this particular factorization $\mathbf{P} = \mathbf{U} \mathbf{U}^T$ before. I have a couple of questions:
- Is it still, by definition, Cholesky factoriation? If not, what is it called?
- Is the simple means to compute this particular variant (e.g. a MATLAB command)
- Is there a specific relationship between $\mathbf{U}$ and $\mathbf{R}$?