Exercise 6.5.2

Just follow the process of diagonalization. But remember that if the dimension of some eigenspace is more than $1$, we should choose an orthonormal basis on it.

1.

$$P=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right),D=\left(\begin{array}{cc}\hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right).$$

2.

$$P=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill i\hfill & \hfill -i\hfill \end{array}\right),D=\left(\begin{array}{cc}\hfill -i\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill i\hfill \end{array}\right).$$

3.

$$P=\frac{1}{\sqrt{3}}\left(\begin{array}{cc}\hfill 1\hfill & \hfill \sqrt{2}\hfill \\ \hfill i+1\hfill & \hfill -\frac{i+1}{\sqrt{2}}\hfill \end{array}\right),D=\left(\begin{array}{cc}\hfill 8\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right).$$

4.

$$P=\left(\begin{array}{ccc}\hfill \frac{1}{\sqrt{3}}\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill \frac{1}{\sqrt{6}}\hfill \\ \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{\sqrt{3}}\hfill \\ \hfill \frac{1}{\sqrt{3}}\hfill & \hfill -\frac{1}{\sqrt{2}}\hfill & \hfill \frac{1}{\sqrt{6}}\hfill \end{array}\right),D=\left(\begin{array}{ccc}\hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -2\hfill \end{array}\right).$$

5.

$$P=\left(\begin{array}{ccc}\hfill \frac{1}{\sqrt{3}}\hfill & \hfill \frac{1}{\sqrt{2}}\hfill & \hfill \frac{1}{\sqrt{6}}\hfill \\ \hfill \frac{1}{\sqrt{3}}\hfill & \hfill 0\hfill & \hfill -\frac{\sqrt{2}}{\sqrt{3}}\hfill \\ \hfill \frac{1}{\sqrt{3}}\hfill & \hfill -\frac{1}{\sqrt{2}}\hfill & \hfill \frac{1}{\sqrt{6}}\hfill \end{array}\right),D=\left(\begin{array}{ccc}\hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$