Why is $\left( {\begin{array}{*{20}{c}} B & B \\ B & B \\ \end{array}} \right)$ positive semidefinite? Let $B\in M_n$ be positive semidefinite .
Why is $\left( {\begin{array}{*{20}{c}}
   B & B  \\
   B & B  \\
\end{array}} \right)$ positive semidefinite?

Let $x = \left( \begin{matrix} x_1 \\ x_2 \end{matrix} \right) \in \mathbb{C}^{2n}$ with $x_1 ,x_2 \in \mathbb{C}^n$. Then we have 
\begin{align}
 \langle x, \left( \begin{matrix} B & B \\ B & B \end{matrix} \right)x \rangle &= \langle \left( \begin{matrix} x_1 \\ x_2 \end{matrix} \right) , \left( \begin{matrix} Bx_1 + Bx_2 \\ Bx_1 + Bx_2 \end{matrix}\right) \rangle \\
&= \langle x_1 , Bx_1 +Bx_2\rangle + \langle x_2, Bx_1 + Bx_2 \rangle \\
&= \langle x_1 + x_2 , B(x_1 + x_2 ) \rangle \\
&\ge 0
\end{align}
As $x_1 + x_2 \in \mathbb{C}^n$ and $B$ is positive semidefinite.
Also, $$\left( \begin{matrix} B & B \\ B& B \end{matrix} \right)^\dagger = \left( \begin{matrix} B^\dagger & B^\dagger \\ B^\dagger & B^\dagger \end{matrix} \right) = \left( \begin{matrix} B & B \\ B& B \end{matrix} \right).$$
So $\left( \begin{matrix} B & B \\ B& B \end{matrix} \right)$ is positive semidefinite.
