Are positive definite matrices necessarily diagonalizable and when does the famous eigenvalue criterium apply?

Are positive definite matrices necessarily diagonalizable and when does
the famous eigenvalue criterium apply?

I mean in $\mathbb{C}$ positiv definite matrices seem to be self-adjoint.
For matrices over real vector spaces this seems to be wrong, but is it
still true that they are diagonalizable?
Then everyone knows a result similar to this: When a matrix has only
positive eigenvalues then it is positive definite. Is this result always
true and do we also have the converse in general?