Exercise 6.2.3

Carry out elimination on matrix $M$, we have:

$M=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 4\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 4\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill -1\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 4\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{13}{4}\hfill \end{array}\right]$

Since $S=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right]$, and $A=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]$, we have

$-A^T{S}^{-1}A=-\left[\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]=-\frac{13}{4}$

So it matches the block elimination as well.

There are 2 positive pivots for $M$ and 2 positive eigenvalues for $M$