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 $\mathbf{R}$?