For the rotation matrix \begin{bmatrix} cos(t) & sin(t) & 0 \\ -sin(t) & cos(t) & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} I already got to the three corresponding eigenvalues [1, cos(t) + i sin(t), cos(t) - i sin(t)].
Now I don't know how to apply them to find each corresponding eigenvectors. I tried some examples I saw in YouTube, but they are only examples about matrices with real numbers, and in this case the eigenvectors involve complex numbers.
In a solutions book I found that the eigenvectors are:
For e-val1 = |0 0 1|
for e-val2 = |$\frac{1}{\sqrt{2}}$ $\frac{i}{\sqrt{2}}$ 0|
and for e-val3 = |$\frac{1}{\sqrt{2}}$ $\frac{-i}{\sqrt{2}}$ 0|
But I have no idea on how to get to those values. The examples I saw simply don't work.
Could someone please, if it's not trouble, tell me some tip, clue or explanation on how to reach those eigenvectors? Maybe I'm missing some trigonometric identity or something like that.