Checking psd-ness of matrix I have the following problem and don't know how to proceed...
I want to check if
\begin{equation}
\frac{1}{2}(B^\top A^\top A + A^\top A B) - \frac{1}{4}B^\top A^\top A A^\top (AA^\top AA^\top)^{-1}AA^\top A B
\end{equation}
is positive semi-definite (psd). We have $A \in \mathbb{R}^{m \times n}$ with full row rank. In addition, $B \in \mathbb{R}^{n \times n}$ is psd and symmetric. Could anyone please give me a hint?

I don't finished the problem, but maybe these equalities can to help you.
Note that:
$$B^T A^T A A^T (AA^T AA^T)^{-1}AA^T AB=B^T A^T A A^T \left((A^T)^{-1}A^{-1}(A^T)^{-1}A^{-1}\right) A A^T AB$$
Then,
$$B^T A^T A A^T (AA^T AA^T)^{-1}AA^T AB=B^T A^T AB=(AB)^T AB.$$
And
$$B^T A^T A + A^T AB=(AB)^T A + A^T AB. $$
