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It is no doubt that R has a Cholesky decomposition when R is a positive definite matrix.I want to ask Whether R has a Cholesky decomposition when R is a positive semi-definite?I would appreciate it if you could give an example.Thank you.

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The answer is "yes", you can find a non-constructive proof of this fact also in Wikipedia:

http://en.wikipedia.org/wiki/Cholesky_decomposition#Proof_for_positive_semi-definite_matrices

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  • $\begingroup$ Thank you for your answer.But when R is a positive definite matrix,is there any method to do a Cholesky decomposition for R?For example,in the tool of MATLAB,we know that the instruction of chol can do a Cholesky decomposition when R is a positive definite matrix. $\endgroup$
    – Purple
    Commented Jan 22, 2014 at 14:19
  • $\begingroup$ Yes, of course there is a method in the positive definite case, see here: en.wikipedia.org/wiki/Cholesky_decomposition#Computation $\endgroup$
    – Paglia
    Commented Jan 22, 2014 at 16:26
  • $\begingroup$ In Matlab Programming, the "chol" command can be used to simply apply this to only a positive definite matrix.When it comes to a positive semi-definite matrix，“chol” fails.I want to ask if there any command in Matlab Programming that can be used to a positive semi-definite matrix? $\endgroup$
    – Purple
    Commented Jan 23, 2014 at 0:46

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