It’s from the book “linear algebra and its application” by gilbert strang, page 260.
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?
A matrix A has an eigenvalue λ if and only if A−1 has eigenvalue λ−1. To see this, note that
If your matrix A has eigenvalue λ, then I−A has eigenvalue 1−λ and therefore (I−A)−1 has eigenvalue 11−λ.