# Cholesky decomposition of positive semidefinite matrix

Dear experts,

for a Monte Carlo study I need a Cholesky decomposition of a correlation matrix that is not positive definite, but positive-semidefinite. As I found out, one way is adjusting the matrix, an other way adjusting the method of computing the cholesky decomp. Unfortunately all algorithms I know to adjust a matrix only produce semi-definite matrices. Do you have a tip for me?

Thanks,

Albert
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Commented:
This a nasty problem to be dealing with. Cholesky decompositions of semi definite matrices are not unique, although do exist, and standard algorithms do not work.

One way of getting a solution is to multiple the diagonal elements by 1 + k where k>0 is really small, this will result in a +ve definite matrix.  Solving by standard means, for different values of small k>0, you can extrapolate the matrix values back to k=0. This will give one of the solutions, assuming convergence as k->0. You can then multiply the resulting matrices to see how well it corresponds to the original for error analysis.
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Author Commented:
Dear GwynforWeb,

this sounds reasonable. I have read there is an alternative method to Cholesky decomposition which uses eigenvalues. Can you recommend this method?

Sincerely,

Albert
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Commented:
Albert - I am not familiar with the eigenvalue technique. I don't envy you working with semi definite matrices , there are no really definitive algorithms or code around for dealing with them. I'd use what suits you best and is implementable in reasonable time. I'd play around with them and see what works, using the simplest first. The 'hack' I presented might well give sufficient accuracy for a single k = 10e-10 for instance.
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Author Commented:
Thank you very much. I will try out all possibilities and see what works.

Sincerely,

Albert
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