The product of two positive definite matrices has real and positive eigenvalues? [duplicate]
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.