Symmetric positive semidefinite closedness

Question

Given $A,B$ symmetric positive semidefinite matrices of equal size it is clear $A^2,B^2,(A+B)^2,A+B$ are all symmetric positive semidefinite.

However is $2AB$ symmetric positive semidefinite?

Answer 1

$2AB$ doesn't even need to be symmetric - much less symmetric and positive semidefinite - given two symmetric, positive semi-definite matrices $A, B$. Let us take the following counterexample:

$$A = \pmatrix{ 1 & 1 \cr 1 & 2 \cr}, B = \pmatrix{2 & 1 \cr 1 & 1}, AB = \pmatrix{ 3 & 2 \cr 4 & 3 \cr}$$

Both $A$ and $B$ are symmetric, positive, semi-definite matrices, but $AB$ isn't symmetric.