Eigenvalues of $I_n-A$

Question

Is there a simple relationship betweeen the eigenvalues of a $n\times n$ matrix $A$ and the matrix $(I_n-A)$? I beg your pardon if this questions has already been answered.

Answer 1

Yes there is. If $\lambda$ is an eigenvalue of $A$, then $1-\lambda$ is an eigenvalue of $I-A$. In general for any polynomial $p$, $p(\lambda)$ is an eigenvalue of $p(A)$. Additionally, $A$ and $p(A)$ has the same eigenvectors.

Answer 2

Suppose $Av= \lambda v$. Then $(I_n -A)v = v - \lambda v = (1-\lambda) v$.

So the spectrum of $I_n-A$ is $\{ 1- \lambda : \lambda \mbox{ eigenvalue of $A$} \}$