We have $A \in \mathbb{R}^{n \times n}$ which is symmetric and positive-definite. Also, $A$ is a block matrix:
$$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{pmatrix}$$
Let, $S = A_{22} - A_{21}A_{11}^{-1}A_{12}$ be the Schur complement. How can one demonstrate that,
$$||S||_{2} \le ||A||_{2}?$$ I looked at the block Cholesky factorization given in this document but it is no help. Thanks!
https://www.cs.cornell.edu/~bindel/class/cs4220-s16/lec/2016-02-17-notes.pdf