How to prove product of positive definite matrices are diagonalizable?

Question

I'm trying to solve a problem and need to prove that product of two positive definite matrices is diagonalizable.

I know matrices are diagonalizable when they are symmetric. How do I prove that product of 2 positive definite matrices is symmetric?

thanks!

Answer 1

If $A,B$ are positive definite and $A^{1/2}$ denotes the (unique) positive definite square root of $A$, then $AB$ is similar to $A^{1/2}BA^{1/2}$, which is symmetric. Because $A^{1/2}BA^{1/2}$ is diagonalizable, $AB$ must be diagonalizable as well.