Homework question.
Let $A$ be a $2 \times 2$ matrix with real entries. Suppose that $A$ has an eigenvalue $\lambda$ with the imaginary part of $\lambda \neq 0$. Is there an orthonormal basis of $\mathbb{C}_2$ consisting entirely of eigenvectors of $A$? Explain.
So far I've got the following:
A matrix is self-adjoint iff its eigenvalues are all real. By the spectral theorem, a linear transformation $T$ from $V$ to $V$ is normal iff its eigenvectors span $V$. A linear transformation is normal iff it commutes with its adjoint.
With these three facts, I can state that A is not self-adjoint, and I need to show that it is not normal, assuming that the proposition is false (my belief).
I've found a couple simple counterexamples, but I can't figure out how to make the jump from not self-adjoint to not-normal. (Or any other method of showing it's not normal).