Exercise 1.10.4

$S-\mathit{\lambda M}=\left[\begin{array}{cc}\hfill 5-\lambda \hfill & \hfill 4\hfill \\ \hfill 4\hfill & \hfill 5\hfill \end{array}\right]$, solve for $\mathit{det}(S-\mathit{\lambda M})=0$, we find ${\lambda }_1=\frac{9}{5}$, the corresponding eigenvector is $x_1=\left[\begin{array}{c}\hfill 5\hfill \\ \hfill -4\hfill \end{array}\right]$.

The other eigenvalue is infinite (why? TODO), so according to equation (10), we have $\mathit{Mx}=0$ when $\lambda =\infty$, solve for this we have: $x_2=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]$.

It’s easy to see that $x_1$ and $x_2$ are $M-$orthogonal to each other, i.e. $x_1^{T}M{x}_2=0$.