|Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices|
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On Wed, 24 Feb 2010, Ben Goodrich wrote:
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
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
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
I don't think it's essential that the matrix is singular; I guess that a
similar example can be constructed for invertible matrices.