Let $A=\begin{bmatrix} A_{11} & A_{21}^T\\ A_{21} & A_{22} \end{bmatrix}\in \mathbb{R}^{nxn}$ , which is a symmetric positive definite matrix, and $A_{11} \in \mathbb{R}^{pxp}$ which is invertible.
(a) Prove the Schur complement
$S=A_{22}-A_{21}A_{11}^{-1}A_{21}^T$
satisfies $\kappa_2(S) \leq \kappa_{2}(A)$
(b) Prove that $\lVert A_{21}A_{11}^{-1} \rVert_2 \leq \kappa_2(A)^{1/2}$
where $\kappa_2$ is the condition number with matrix 2 norm
For problem (a) I have already done it.
I can't figure out how to solve problem (b) though.
I know that in this case $\kappa_2(A)^{1/2}=\frac{\lambda_{max}(A)}{\lambda_{min}(A)}$, and I tried everything in my head but still couldn't get it. Am I missing something?