Parts of a positive semidefinite

Question

I have the following positive semidefinite matrix \begin{align} L=\left [ \begin{matrix} A & \tau \\ \tau^T & p\\ \end{matrix} \right ]>=0 \end{align} What can be said about A ? Where $\tau $ is a vector and $p$ is a scalar.

Answer 1

$A$ itself must be positive semidefinite. Just consider $$\begin{pmatrix}v^t & 0\end{pmatrix} \begin{pmatrix} A & \tau \\ \tau^t & p \end{pmatrix} \begin{pmatrix}v \\ 0 \end{pmatrix}.$$ In general you'll find that $p\geq 0$ and $p A - \tau \cdot \tau^t \geq 0$.