# Eigenvalues of the product of symmetric and positive definite matrices

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?

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.
• I'm probably being dense here, but how did you go from $S^t S B$ to $S^t B S$?
• @Bungo by typing too fast, it's $SBS^t.$ Apr 27, 2021 at 21:26