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.

Depends on your definition of positive-definite matrices. Usually they're assumed to be symmetric, and so all their eigenvalues are real. In fact, since they're positive-definite, all their eigenvalues are positive: if $v$ is an eigenvector with eigenvalue $\lambda$ of the positive-definite matrix $A$ then $0 < v'Av = \lambda \|v\|^2$.
