Exercise 1.7.5

$S$ has eigenvalues of ${\lambda }_1=4$ and ${\lambda }_2=2$, with corresponding eigenvectors $x_1=\frac{1}{\sqrt{2}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$ and $x_2=\frac{1}{\sqrt{2}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$. So we have

$B$ has eigenvalues of ${\lambda }_1=0$ and ${\lambda }_2=25$, with corresponding eigenvectors $x_1=\frac{1}{5}\left[\begin{array}{c}\hfill -4\hfill \\ \hfill 3\hfill \end{array}\right]$ and $x_2=\frac{1}{5}\left[\begin{array}{c}\hfill 3\hfill \\ \hfill 4\hfill \end{array}\right]$. So we have