Q:

Is there any relation between the principal eigenvector of the original matrix and its inverse?

This question pop'd up when I was studying graph.
I am thinking about the relation between principal eigenvector of adjacency matrix $A$ and its inverse $A^{-1}$, do they have any relation?

A:

Suppose that $v$ is an eigenvector of $A$ with eigenvalue $\lambda.$ Then $v$ is also an eigenvector of $A^{-1}$, but with eigenvalue $1/\lambda:$
$$v = 1.v = (A^{-1}A)v = A^{-1}(\lambda v) = \lambda (A^{-1}v) \quad \Rightarrow \quad A^{-1}v = \lambda^{-1} v.$$
Now let $\{ \lambda_i \}_{i=1}^n$ be the spectrum of $A$, and let the $\lambda_i$ be ordered: $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n.$ What can you say about the spectrum of $A^{-1}$?

