So, I was trying to construct an operator for an orthonormal vector space $\mathbb{C}^2$. Suppose, I have two orthonormal vectors, $|x\rangle, |y\rangle$ such that $$\langle x|x\rangle=1,\langle y|y\rangle=1,\langle x|y\rangle=0.$$
Then, let's assume I have an operator $A$ such that $$A|x\rangle=\lambda_1|x\rangle,\quad A|y\rangle=\lambda_2 |y\rangle \\ \text{Then}, \langle x|A|y\rangle=\langle x|A|A|y\rangle = 0\\ \rightarrow\left[A\right]=\left[A\right]\left[A\right]$$
So, I should be looking for a 2X2 square matrix $A$ which is equal to it's square? From my knowledge, only an identity matrix or a zero matrix would work in this circumstance, but shouldn't this matrix be a Hermitian, since the operator necessarily needs to be a Hermitian, if it needs to have orthogonal eigenvectors? Am I going wrong somewhere?