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You might want to use the fact that: $$ ||A||_2=\sigma_\max(A) $$ where $\sigma_\max$ is the largest singular value. If you are interested in details, this Math SO question should be interesting. Thus, $$ ||A^{-1}||_2=\frac{1}{\sigma_\min(A)} $$ where $\sigma_\min$ is the smallest singular value. You certainly want to avoid the actual calculation of the ...

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