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