Exercise 1.7.3

• $S=\left[\begin{array}{cc}\hfill 1\hfill & \hfill b\hfill \\ \hfill b\hfill & \hfill 9\hfill \end{array}\right]$, when $9-b^2>0$, i.e. $-3<b<3$, $S$ is p.d.

$S=\mathit{LD}{L}^T=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill b\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 9-b^2\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill b\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

• $S=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 4\hfill & \hfill c\hfill \end{array}\right]$, when $2c-16>0$, i.e. $c>8$, $S$ is p.d.

$S=\mathit{LD}{L}^T=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill c-8\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

• $S=\left[\begin{array}{cc}\hfill c\hfill & \hfill b\hfill \\ \hfill b\hfill & \hfill c\hfill \end{array}\right]$, when $c>0$ and $c^2-b^2>0$, i.e. $-c<b<c$, $S$ is p.d.

$S=\mathit{LD}{L}^T=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill \frac{b}{c}\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill c-\frac{b^2}{c}\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill \frac{b}{c}\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$