Positive semidefinite Matrix condition for Off-Diagonals Suppose I have a matrix of the shape $R=\begin{pmatrix}
A & C^T\\
C & B \end{pmatrix} \in \mathbb{R}^{D \times D}$. The submatrices $A \in \mathbb{R}^{D/2 \times D/2}$ and $B \in \mathbb{R}^{D/2 \times D/2}$ are known and are PSD.
I wonder what structure $C$ needs to have, such that the  overall matrix $R$ is also PSD.
My attempt:
The correlation of two variables is defined as : $\rho(x,y)=\frac{cov(x,y)}{\sigma_x \sigma_y}$. I thought as long as the entries in $C$ follow this condition, i.e. any entry $|C_{i,j}|^2 \leq diag(A)_i diag(B)_j$, the resulting matrix $R$ will be PSD. However I sampled some random correlation martrices and this seems not to hold.

The lowest value of the spectrum of a self-adjoint matrix $S$ is achieved by some unit vector as $\langle x,Sx\rangle$.
Therefore the requirement that $S=\begin{pmatrix}A&C^\top\\C&B\end{pmatrix}$ is positive becomes $$0\le(x,y)\begin{pmatrix}A&C^\top\\C&B\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\langle x,Ax\rangle+\langle y,By\rangle+2\langle y,Cx\rangle$$ Hence
$$2|\langle y,Cx\rangle|/\|x\|\|y\|\le \langle x,Ax\rangle\|x\|/\|y\|+\langle y,By\rangle\|y\|/\|x\|$$ equivalently, for any unit $x,y$, $$|\langle y,Cx\rangle|\le \sqrt{\langle x,Ax\rangle\langle y,By\rangle}$$ A sufficient condition is
$$\forall x,y,\quad |\langle y,Cx\rangle|\le \sqrt{ab}$$ where $a,b$ are the minimum eigenvalues of $A$ and $B$ respectively. In particular, if each coefficient of $C$ satisfies this condition.
