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?


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