Hermitian matrix and eigenvalues

Question

It is true that: Every eigenvalue of a Hermitian matrix is real.

But does this mean that : if all of the eigenvalues of a matrix is real, then the matrix is Hermitian?

Answer 1

NO. Take, for instance $$A=\begin{pmatrix}1&i\\0&2\end{pmatrix}$$

Answer 2

Any real nonsymmetric matrix is not Hermitian. For example, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} And eigenvalues are only 0. (This does not mean any real valued matrix have only real eigenvalues, I mean we can find very simple examples instead of using complex number containing $i$)