Eigenvalues of $A +A^{-1}$

Question

Knowing the eigenvalues of $A$, what we can say about the eigenvalues of $A+A^{-1}$?

Is it true that $\lambda_{i,(A+A^{-1})} = \lambda_{i,A} + \frac{1}{\lambda_{i,A}}$, where $\lambda_{i,A}$ is the $i$-th eigenvalue of $A$?

I ask this question after asking this question, due to the different opinion in the conversation in the comment.

Answer 1

Consider a Schur decomposition of $A$: $A = UTU^{*}$, where $T$ is upper triangular with eigenvalues $\lambda_j(A)$ on the diagonal and $U$ is unitary matrix. Clear that $A^{-1} = U T^{-1} U^{*}$, where $T^{-1}$ is again an upper triangular matrix with eigenvalues $1/\lambda_j(A)$ of $A^{-1}$ on the diagonal. Then $A + A^{-1} = U (T + T^{-1}) U^{*}$ which implies that eigenvalues of $A + A^{-1}$ are exactly $\lambda_j(A) + 1/\lambda_j(A)$