Exercise 2.1.4

Check matrix $M={(-S+2I)}^{-1}{S}^T$.

• For $n=2$, we have $S=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$, $S^T=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$, $-S+2I=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 2\hfill \end{array}\right]$, so ${(-S+2I)}^{-1}=\left[\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill \end{array}\right]$ and $M=(-S+2I)^{-1}ST =

  $\left[\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill \end{array}\right]$ $\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$

  =

  $\left[\begin{array}{cc}\hfill 0\hfill & \hfill \frac{1}{2}\hfill \\ \hfill 0\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]$

  $. The matrix $M$ has eigenvalues ${\lambda }_1=0,{\lambda }_2=\frac{1}{4}$.

• For $n=3$, we have $S=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$, $S^T=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$, $-S+2I=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill \end{array}\right]$, so ${(-S+2I)}^{-1}=\left[\begin{array}{ccc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill \frac{1}{8}\hfill & \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill \end{array}\right]$ and $M=(-S+2I)^{-1}ST =

  $\left[\begin{array}{ccc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill \frac{1}{8}\hfill & \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill \end{array}\right]$ $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$

  =

  $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{4}\hfill & \hfill \frac{1}{2}\hfill \\ \hfill 0\hfill & \hfill \frac{1}{8}\hfill & \hfill \frac{1}{4}\hfill \end{array}\right]$

  $. The matrix $M$ has eigenvalues ${\lambda }_1=0,{\lambda }_{2,3}=\frac{1}{4}$.