I have the following question:
Let $V=\mathbb{C}³$ with canonical inner product and let $T\in \mathcal{L}(V)$ satisfing the following properties:
i) The only eigenvalues of $T$ are $i$ and $-i$
ii) $T$ is normal
iii) The eigenvectors of $i$ are $V_i=[(i,1,1),(i,0,1)]$
Then what is $T(1,2,i)$?
The answer should be $(i,2,1),(0,i,2),(-i,2i,1),(1,2,3i)$ or $(1,2i,1)$
I found a eigenvector of $-i$ by finding the orthogonal complement of $V_i$ and then the coordinates of $(1,2,i)$ in the basis of eigenvectors and then just use the matrix of $T$ in the diagonalized form and take the appropiated change of basis... but I didn't used that $T$ is normal and the result isn't right.
Anyone could help me?