Cholesky decomposition vs. $\mathbf L\mathbf D\mathbf L^\top$ decomposition In different books and on Wikipedia, you can see frequent mentions of Cholesky decomposition and $\mathbf L\mathbf D\mathbf L^\top$ decomposition is seldom mentioned. Why so? As far as I understand, $\mathbf L\mathbf D\mathbf L^\top$ decomposition can be applied to a broader range of matrices (we don't need a matrix to be positive-definite) and for the same computational price. 

A frequent reason for preferring $\mathbf L\mathbf D\mathbf L^\top$ over Cholesky is that the latter requires the evaluation of square roots, while the former does not. In symbolics, where one might want to keep the entries radical-free, or in numerics, where the square root operation on a particular machine might be slower than division, this can be a deciding factor.
As for positive-definiteness: it is true that $\mathbf L\mathbf D\mathbf L^\top$ can work on matrices that aren't positive definite, but it is no longer guaranteed to be numerically stable. Also, without an extra operation like pivoting, $\mathbf L\mathbf D\mathbf L^\top$ will fail on matrices such as
$$\begin{pmatrix}0&1\\1&0\end{pmatrix}$$
or in general, matrices with a singular leading submatrix.
(This got too long for a comment.)
