If eigenvectors of different eigenvalues are orthogonal, then is the matrix normal?

Question

It is axiomatic that eigenvectors of normal matrices are orthogonal. Then how about the converse? Is it still true?

Answer 1

Counterexample: the matrix $$\begin{bmatrix}0&0&0\\0&1&1\\0&0&1\end{bmatrix}$$ meets your criteria, but isn’t even diagonalizable.