Positive definiteness of $(\boldsymbol{A} + \boldsymbol{B})(\boldsymbol{A} + \boldsymbol{B})^\top$.

Question

Let $\boldsymbol{A},\boldsymbol{B}$ both be $m \times n$ matrices with $m < n$. Suppose it is known that $\boldsymbol{A}\boldsymbol{A}^\top$ is positive definite and that $\boldsymbol{B}\boldsymbol{B}^\top$ is not. Will $(\boldsymbol{A} + \boldsymbol{B})(\boldsymbol{A} + \boldsymbol{B})^\top$ be positive definite?

Answer 1

It need not be, for example, let $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ \end{bmatrix}$ then $AA^T=I_2$

Now let $B=\begin{bmatrix} -1 & 0 & 0 \\ 0 & 0 & 0\\ \end{bmatrix}$, then $BB^T$ is not positive definite.

and $A+B$ is of rank $1$, hence $(A+B)(A+B)^T$ is not positive definite.