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I have a square matrix $A$.

Is it possible to determine if its largest eigenvalue is smaller (by magnitude) than 1 by inspecting the matrix $(I-A)^{-1}$?

(we can assume that $I-A$ is invertible.)

EDIT: My question is quite simple, if not more specific. Obviously you could recover $A$ and compute its spectrum. But I am looking for something which is algorithmically simple...

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