I have this question:

Let $\{u_1, u_2, u_3, u_4\}$ be a basis for $\mathbb{R}^4$ and $B$$4 \times 4$ matrix such that: $$ 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.
Justify your answers.
[image of problem statement]

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.

  • 2
    $\begingroup$ Try $u_1+u_2$ . $\endgroup$
    – Eric
    Apr 25, 2021 at 4:41
  • 1
    $\begingroup$ Welcome to MSE. Please type your questions rather than posting images. Images can't be browsed, and are not accessible to those using screen readers. If you need help formatting math on this site, here's a tutorial To begin with, surround all math expressions (including numbers,) with $ signs. Use ^ for exponents and _ for subscripts. $x_1^{2/3}$ shows up as $x_1^{2/3}$. $\endgroup$
    – saulspatz
    Apr 25, 2021 at 4:50
  • $\begingroup$ You may want to check eigenvalues of a permutation matrix. $\endgroup$
    – Toni Mhax
    Apr 26, 2021 at 6:02

1 Answer 1


Relative to the given basis, the matrix is $M=\begin {pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix} $.

Now find the roots of the characteristic polynomial: $\rm {det}(M-xI) $.

Get $x^2(x^2-1)+(1-x^2)=x^4-2x^2+1=(x+1)^2 (x-1)^2$.

Thus the eigenvalues are $\pm1$.

  • $\begingroup$ Sorry, how did you derive matrix M? $\endgroup$
    – user877841
    Apr 26, 2021 at 6:28
  • $\begingroup$ Applied the transformation to the basis vectors, expressed in terms of themselves. Which is to say to the standard basis. Then expressed the results in terms of that same basis. $\endgroup$
    – user403337
    Apr 26, 2021 at 6:34

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