Exercise 5.2.2

For these question, see the direction of the subsection “Test for Diagonalization”.

1.

It’s not diagonalizable since $\dim <!--\; FUNCTION\; APPLICATION\; -->(E_1)$ is $1$ but not $2$.

2.

It’s diagonalizable with $D=\left(\begin{array}{cc}\hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill \end{array}\right)$ and $Q=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right)$.

3.

It’s diagonalizable with $D=\left(\begin{array}{cc}\hfill -2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 5\hfill \end{array}\right)$ and $Q=\left(\begin{array}{cc}\hfill 4\hfill & \hfill 1\hfill \\ \hfill -3\hfill & \hfill 1\hfill \end{array}\right)$.

4.

It’s diagonalizable with $D=\left(\begin{array}{ccc}\hfill 3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right)$ and $Q=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right)$.

5.

It’s not diagonalizable since its characteristic polynomial does not split.

6.

It’s not diagonalizable since $\dim <!--\; FUNCTION\; APPLICATION\; -->(E_1)$ is $1$ but not $2$.

7.

It’s diagonalizable with $D=\left(\begin{array}{ccc}\hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right)$ and $Q=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill -1\hfill & \hfill -1\hfill \end{array}\right)$.