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A diagonal-pivot LDL decomposition of a real symmetric matrix has the formwhere $L$ is a unit lower triangular matrix and $D$ is a diagonal matrix. When the needed scalar pivots are nonzero, symmetric Gaussian elimination computes one Schur complement at a time:In particular, nonzero leading principal minors guarantee this form, and a symmetric positive-definite matrix has positive pivots. Conversely, $d_j>0$ for every $j$ gives $x^{T}Ax=(L^{T}x{)}^{T}D(L^{T}x)>0$ for every nonzero vector $x$.

For $A=\left(\begin{array}{cc}4& 2\\ 2& 3\end{array}\right)$, take $L=\left(\begin{array}{cc}1& 0\\ 1/2& 1\end{array}\right)$ and $D=\mathrm{diag}(4,2)$. General symmetric matrices need more care: $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ has no such unit-lower-triangular/diagonal factorization. Symmetric indefinite numerical factorization can instead use permutations and symmetric diagonal blocks of sizes one or two.^[ref]