Question: Suppose is an eigenvalue of the B in Exercise, and that x is a corresponding eigenvector, so that (A - αI)-1x = μx. Use this equation to find an eigenvalue of A in terms of μ and α.
Exercise: Suppose Ax = λx with x ≠ 0. Let ? be a scalar different from the eigenvalues of A, and let B = (A - αI)-1. Subtract αx from both sides of the equation Ax = λx, and use algebra to show that 1/(λ - α) is an eigenvalue of B, with x a corresponding eigenvector.