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

[Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx>]

On Wed, 24 Feb 2010, Gael Guennebaud wrote:

> note that the LDLT decomp does pivoting, so it is normal that you don't 
> get the same diagonal matrix. If you reconstruct the matrix from the 
> decomp you will see that in this case the decomp is correct.

To further clarify, because this also threw me off for a while: I think 
the important thing here is that the matrix A is singular. The diagonal D 
is unique (up to reordering) for positive-definite matrices. However, 
there is no such uniqueness for singular matrices. Simple example

A = [ 0  0;  0  1 ]      (these are 2-by-2 matrices in Matlab notation)
L = [ 0  0;  sqrt(3)  1 ]
D = [ 1/4  0;  0  1/4 ]

The matrix A has two LDL^T decompositions, A = A * I * A^T = L * D * L^T, 
but the diagonal parts are not the same.

Cheers,
Jitse