[Math] If a matrix has positive, real eigenvalues, is it always symmetric

Question

We know that symmetric matrices are orthogonally diagonalizable and have real eigenvalues. Is the converse true? Does a matrix with real eigenvalues have to be symmetric?

A class of symmetric matrices, the positive definite matrices, have positive real eigenvalues. Is the converse true? Does a matrix with positive real eigenvalues have to be symmetric, positive-definite?

I think the answer to all this is "no", but I just wanted to confirm.

Thanks,

Answer 1

Is the matrix

$$\begin{pmatrix}1&1\\0&1\end{pmatrix}$$

symmetric? It has only one positive eigenvalue of multiplicity two.