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Let V be a finite-dimensional inner product space over $K=\mathbb{C}$ and $T\in L(V)$. Let $\mathbb{C}^2 \to \mathbb{C}^2$ be defined by $T(z_1, z_2) = (iz_2, 4iz_1).$ Prove it is possible to have v be an eigenvector of T with eigenvalue $\lambda$ but v is not an eigenvector of T* with eigenvalue $\overline{\lambda}$.

I found $\lambda = 2i, -2i.$ I understand what I have to prove, but I'm having some difficulties proving it. Any help would be appreciated.

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Take, for instance, $v=(1,2)$. It is an eigenvector of $T$, with eigenvalue $2i$. But it is not an eigenvector of $T^*$.

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