Orthogonal diagonalisation of a $4\times 4$ matrix Can somebody help me to orthogonally diagonalise the matrix $\begin{bmatrix}0 & 0 & 0 & 1\\0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{bmatrix}$?

An Eigenvector $(v_1,v_2,v_3,v_4)$ associated to an eigenvalue $\lambda$ satisfies $$
\begin{cases}
x_4 =\lambda x_1\\
x_3 = \lambda x_2\\
x_2 = \lambda x_3\\
x_1 =\lambda x_4
\end{cases}$$
Hence $\lambda^2=1$ proving that the eigenvalues are $-1$ and $1$. From there you can easily find an orthonormal basis of eigenvectors.
