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?
A matrix $A$ is positive definite iff $\forall x, x^TAx>0$.
No symmetry is implied here.