I have this question:
Let $\{u_1, u_2, u_3, u_4\}$ be a basis for $\mathbb{R}^4$ and $B$ a $4 \times 4$ matrix such that: $$ Bu_1=u_2,~~ Bu_2=u_1,~~ Bu_3=u_4,~~ Bu_4=u_3.~~ $$ Find all eigenvalues of $B$ and determine whether $B$ is diagonalizable.[image of problem statement]
Justify your answers.
I am having troubles identifying all the eigenvalues of the matrix $B$ described in this question.
I believe that one of the eigenvalues is $-1$, which can be obtained from $$\begin{eqnarray} Bu_1 - Bu_2 &=& u_2 - u_1 \\ \Rightarrow~~ B(u_1 -u_2) &=& u_2 - u_1 \end{eqnarray}$$
The above can be true only if $\lambda = -1$. This is the same for the $Bu_3 - Bu_4 = u_4 - u_3$
I can't seem to determine the rest or prove that this is the only eigenvalue.
$
signs. Use ^ for exponents and _ for subscripts.$x_1^{2/3}$
shows up as $x_1^{2/3}$. $\endgroup$