Positive Definite Matrix (Block Matrix) Let $B$ be an $m\times n$ matrix. Is $$ A=\begin{pmatrix} I & B \\ B^T & I+B^TB \end{pmatrix} $$ positive definite?
Attempt:
Let $\mathbf{z}=\begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix}$. To show that $A$ is positive definite, $\mathbf{z}^TA\mathbf{z}>0$. Expanding $\mathbf{z}^TA\mathbf{z}>0$ gives $\mathbf{x}^T\mathbf{x}+\mathbf{y}^T\mathbf{y}+(B\mathbf{y})^T(B\mathbf{y})+2\mathbf{x}^TB\mathbf{y}$. The first three terms are positive, but what can be concluded about the $2\mathbf{x}^TB\mathbf{y}$ term?

One hint could be to try to rewrite with sum of self-outer products:
$$A = \begin{bmatrix}I&B\\B^T&I+B^TB\end{bmatrix}= \cdots\\=\begin{bmatrix}I\\B^T\end{bmatrix} \begin{bmatrix}I&B\end{bmatrix} + \begin{bmatrix}0\\I\end{bmatrix} \begin{bmatrix}0&I\end{bmatrix}$$
What can we say about it now?
