Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices

[Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx>]

On Wed, 24 Feb 2010, Ben Goodrich wrote:

> http://www.pages.drexel.edu/~hvb22/dimacsppr3.pdf
>
> they claim (without proof) that D is unique in the positive
> semi-definite case (although part of L is not when A is singular).
> Maybe they are incorrect, but I don't think your example shows that
> because
>
> L = [ 0  0;  sqrt(3)  1 ]
>
> is not *unit* lower triangular.

You're right. Apologies for the confusion I caused. Let me try again.

It looks like the non-uniqueness of D is caused by the pivoting. Here's 
another example:

L1 = [ 1 0; 0.5 1 ]
D1 = [ 4 0; 0 0 ]

L2 = [ 2 1; 1 0 ]
D2 = [ 1 0; 0 0 ]

Now L1 * D1 * L1^T = L2 * D2 * L2^T = [ 4 2; 2 1 ]. Both are valid LDL^T
decompositions with pivoting: L1 is unit lower triangular, and L2 is unit 
lower triangular after the exchanging the rows. So pivoting can change the 
diagonal matrix.

I don't think it's essential that the matrix is singular; I guess that a 
similar example can be constructed for invertible matrices.

Cheers,
Jitse