Cholesky decomp vs A = L'DL decomp.

[WCMU101]

Hey all. I've written an algorithm to find the Cholesky decomposition of a symmetric, positive-definite matrix (A). I've used the algorithm from: http://en.wikipedia.org/wiki/Cholesky_decomposition#Avoiding_taking_square_roots

Ok here is my question. My current algorithm solves for A = LDL^T, however I would also like to solve for A = L^TDL. I've seen the results of both decompositions and it looks like they are related. From what I've seen it looks like the L returned from L'DL is just L from cholesky inverted about the positively sloped diagonal. And same for the D. So for example:

A =

4 2 2
2 4 2
2 2 4

A = LDL'

L =
1 0 0
0.5 1 0
0.5 0.333333 1

D =
4 0 0
0 3 0
0 0 2.66667

A = L'DL

L =

1.0000 0 0
0.3333 1.0000 0
0.5000 0.5000 1.0000

D =
2.6667 0 0
0 3.0000 0
0 0 4.0000

So my question. How can I relate the L from LDL' decomp with the L from L'DL decomp (mathematically).

Thanks,

Nick.

 

[AlephZero]

Your example matrix is a special case because it also "symmetrical top to bottom" and "left to right".

For a general symmmetric matrix, you can get the L^T D L decomoposition by running the L D L^T "backwards" starting from the bottom right corner and working to the top left, but the result will not be as you described it. There is no obvious connection between the two sets of L and D matrices.

As a simple example, take the matrix

1 1
1 2

 

[WCMU101]

Thanks for that! Exactly what I needed to know.

Nick.