Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 1 / 6D (Numerical Analysis)

A unit lower triangular $L$ is invertible. For $x\ne 0$, $y=L^{T}x\ne 0$, so an LDL decomposition with positive diagonal $D$ givesAlso $A=LD{L}^T$ is symmetric. Hence $A$ is a positive-definite matrix.

For the specified matrix, comparing entries successively gives $d_1=1$, $l_{21}=-1$, $l_{31}=2$, $d_2=3-l_{21}^2{d}_1=2$, andThusThe last pivot is negative, so this particular factorization is outside the positive-diagonal case proved above. Indeed $x=(-2,0,1{)}^T$ has $x^{T}Ax=-1$, and $\det A=-11$. There is no positive-diagonal LDL factorization for the printed matrix. The unrestricted LDL decomposition above is nevertheless valid.