# Prove Stability of Iteration Matrix

Given a positive definite Block-Matrix $H > 0$, where $$H = \begin{pmatrix}H_{11} & H_{12} \\ H_{21} & H_{22} \end{pmatrix},$$ where the iteration Matrix $A$ is given by $$A = \begin{pmatrix}0 & -H_{11}^{-1}H_{12} \\ -H_{22}^{-1}H_{21} & 0 \end{pmatrix}.$$ Show that the iteration matrix $A$ is stable, i.e. $$|\text{eig}(A)|<1.$$

I was thinking about the Gershgorin Circle theorem, but I'm not sure how that could be applied to a block matrix. Any suggestions on what a good approach would be?