Can someone help me understand how the addition and subtraction of matricies affects the eigenvalues? For example $B=A+A^{-1}-2I$ₙ Assume matrix $A$ is diagonalizable and has positive real eigenvalues. I just don't understand how I am supposed to know how the eigenvalues interact without knowing the exact matrices specified in the problem. My textbook doesn't mention anything about this and I'm starting to get confused.
 A: In general saying something about the eigenvalues of the sum of two matrices is really hard, even if they are diagonalizable.
However: if they are simultaneously diagonalizable it is a different story.
Suppose that for matrices $A$ and $C$ we have that both $PAP^{-1}$ and $PCP^{-1}$ are diagonal using the same matrix $P$ then it is easy:
The eigenvalues of $A + C$ equal the eigenvalues of $P(A + C)P^{-1}$ and this matrix in turn equals $PAP^{-1} + PBP^{-1}$ and hence is the sum of two diagonal matrices. Computing these eigenvalues is easy.
So the hard part is: how do we know, for given $A$ and $C$ if such a $P$ exist. There is a really nice theorem that states that if $A$ and $C$ are both diagonalizable and they commute (that is: $AC = CA$) then they are simultaniously diagonalizable.
In your case with $C = A^{-1}$ this condition is satisfied.
EDIT: this older question might shed some light on the 'really nice theorem': Simultaneous Diagonalizability of Multiple Commuting Matrices
A: In general if the eigenvalues of two matrices have no bearing on the eigenvalues of their sum.  The exception would be the case where $v$ is an eigenvector of both matrices.
In the example....
Suppose $v$ is an eigenvector of $A$ with eigenvalue $\lambda$ then $v$ will be an eigenvector of $A^{-1}$ with eigenvalue $\frac {1}{\lambda}$
$Bv = Av + A^{-1}v - 2Iv = \lambda v + \frac {1}{\lambda}v - 2v$
And $v$ is an eigenvector of $B$ with eigenvalue $\lambda + \frac {1}{\lambda} - 2$
