Actually Cholesky can be shown to extend to semidefinite matrices:

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

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.139.8064

Evidently implemented in R.

http://tolstoy.newcastle.edu.au/R/e6/help/09/04/9980.html

Actually Cholesky can be shown to extend to semidefinite matrices:

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

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.139.8064

Evidently implemented in R.

http://tolstoy.newcastle.edu.au/R/e6/help/09/04/9980.html

Meanwhile, covariance matrices are semidefinite, read some background at

http://en.wikipedia.org/wiki/Covariance_matrix