Do positive-definite matrices always have real eigen values? Do positive-definite matrices always have real eigenvalues?
I tried looking for examples of matrices without real eigenvalues (they would have even dimensions). But the examples I tend to see all have zero diagonal entries. So they are not positive definite.
Would anyone have an example of positive-definite matrix without any real eigenvalue? Or it is a property of positive-definite matrices that they always have some real eigenvalues?
EDIT:
A matrix $A$ is positive definite iff $\forall x, x^TAx>0$.
No symmetry is implied here.

The matrix $\begin{pmatrix} 2 & 1 \\ -3 & 1 \end{pmatrix}$ is a counterexample. It is positive-definite since for $(x,y) \neq (0,0)$ we have
$$
\begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} 2 & 1 \\ -3 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = 2x^2 - 2xy + y^2 = x^2 + (x-y)^2 > 0.
$$
On the other hand, its characteristic polynomial is
$$(t-2)(t-1)+3 = t^2-3t+5,$$
whose discriminant $9-20 = -11$ is negative, so both of its roots are complex.
