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    $\begingroup$ You might be surprised to find that CholeskyDecomposition[{{a, b, c}, {b, d, e}, {c, e, f}}] works as expected... $\endgroup$ Commented May 14, 2012 at 15:38
  • $\begingroup$ The root cause of this problem is the issue described in this answer by @Leonid - that's at least the conclusion I drew in my answer below. $\endgroup$ Commented May 14, 2012 at 22:31