It’s from the book “linear algebra and its application” by gilbert strang, page 260.

(I−A)−1=I+A+A2+A3+…

Nonnegative matrix A has the largest eigenvalue λ1<1.

Then, the book says, (I−A)−1 has the same eigenvector, with eigenvalue 11−λ1.

Why? Is there any other formulas between inverse matrix and eigenvalue that I don’t know?

**Answer**

A matrix A has an eigenvalue λ if and only if A−1 has eigenvalue λ−1. To see this, note that

Av=λv⟹A−1Av=λA−1v⟹A−1v=1λv

If your matrix A has eigenvalue λ, then I−A has eigenvalue 1−λ and therefore (I−A)−1 has eigenvalue 11−λ.

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