Eigenvalues of the product of symmetric and positive definite matrices

Question

Let $A$ , $B$ be two real symmetric matrices and $A$ is positive definite. Then show that $AB$ has real eigenvalues.

Symmetric matrices have real eigenvalues and product of two symmetric matrices need not be symmetric. How can the positive definiteness of $A$ be used here to show that the eigenvalues of $AB$ are real?

Answer 1

Let $A = S^t S.$ Then $AB = S^t S B.$ The eigenvalues of $AB$ are thus the same as those of $S^tBS,$ which is a symmetric matrix.