Let $T$ be a matrix of which I know its characteristic values, how can I find $\operatorname{Tr}(T-I)^{-1}$?
I know that the sum of the characteristic values is the trace, but I'm having a problem as I can't understand how I can know the new characteristic values of the new matrix using the old characteristic values and the old matrix.
Hint: if you know all of the eigenvalues (characteristic values) of $T$ then you also know the eigenvalues of $T-I$. Now you need two facts: one, $\lambda$ is an eigenvalue of $S^{-1}$ if and only if $\lambda^{-1}$ is an eigenvalue of $S$, and two, the trace of an operator is the sum of its eigenvalues.