Let $M=\begin{pmatrix}A&B\\B'&C\end{pmatrix}$ be semi-positive-definite, that is, $X'MX\geq 0$. Here $A$ is $m\times m$, $C$ is $n\times n$. Then by checking with $X=\begin{pmatrix}x\\ty\end{pmatrix}$, it is easy to find that $(x'By)^2\leq x'Ax y'Cy$ for any $x\in \Bbb R^m$, $y\in\Bbb R^m$.
However, what is the condition on the "=" holds. How to show that "=" hold if and only if there exists a nonzero $\lambda$ such that $\lambda Ax=By, Cy=\lambda B'x$, or $Ax=\lambda By, \lambda Cy=B'x$.
If $x'Ax\neq 0, y'Cy\neq 0$, then it is easy since, $M$ is semi-positive-definite implies that $A,C$ are both semi-positive-definite. we have $Ax=0, By=0$.
However, if $x'Ax=0$ or $y'Cy=0$, how to justify that there exists a nonzero $\lambda$ such that $\lambda Ax=By, Cy=\lambda B'x$, or $Ax=\lambda By, \lambda Cy=B'x$.