Diagonalizing matrix that has variables

Question

So I have the following matrix that I have to diagonalize

\begin{bmatrix} 0 & 1 & 0 & 0 \\ -x^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -y^2 & 0 \end{bmatrix}

So the eigenvalues of the matrix are: ${\pm ix}$, ${\pm iy}$.

But now I'm having trouble finding the eigenvectors. Specifically, I'm stuck on what to do when I normally try to reduce the matrix to row-reduced form. When plugging in +ix, I can cancel out the second row, but then stuck on what to do with the third and fourth row. Any help would be great!

Answer 1

By observation, you can see that multiplying the first column with $\pm\frac{i}{x}$ will cancel the second column out. The same yields for the third and the fourth column with the factor $\pm\frac{i}{y}$. Therefore your eigenvectors will be:

$$\begin{bmatrix} -\frac{i}{x}\\1\\0\\0\end{bmatrix},\begin{bmatrix} \frac{i}{x}\\1\\0\\0\end{bmatrix},\begin{bmatrix} 0\\0\\-\frac{i}{y}\\1\end{bmatrix},\begin{bmatrix} 0\\0\\\frac{i}{y}\\1\end{bmatrix}$$ Corresponding to the eigenvalues: $(ix,-ix,iy,-iy)$