The product of two positive definite matrices has real and positive eigenvalues?

Question

Given two real positive definite (and therefore, symmetric) matrices $A$ and $B$, are all the eigenvalues of $AB$ real and positive?

• Wikipedia says $AB$ is positive definite if $A$ and $B$ are positive definite and commute, but I don't need $AB$ to be symmetric.
• Between the lines of this question the asking user somehow prove that yes, "the eigenvalues of $AB$ are hence real and strictly positive" but I couldn't understand if that is confirmed in the answer.

Answer 1

If we call $B^{1/2}$ the symmetric matrix such that $B^{1/2}B^{1/2}=B$ (i.e. the standard square root of a positive definite matrix) then $$ AB=AB^{1/2}B^{1/2}=B^{-1/2}(B^{1/2}AB^{1/2})B^{1/2}, $$ that is $AB$ is similar to the positive definite matrix $B^{1/2}AB^{1/2}$, sharing all eigenvalues. It makes the eigenvalues of $AB$ be positive.