Exercise 3.2.9

We have $S=\mathit{LD}{L}^T$, so $D$ is diagonal and thus symmetric, $L$ is lower triangular matrix, which is invertible, so we see that $S$ is a congruent matrix to $D$. Apply the “Law of Inertia”, we can say that $S$ and $D$ have the same number of positive/negative/zero eigenvalues. On the other side, we know that the eigenvalues of $D$ are actually the elements on its diagonal, i.e. the pivots of $S$.

So we conclude that the signs of the pivots of $S$ (in $D$) match the signs of the eigenvalues of $S$.