Thread: Characteristic Value and Vector

If A is nonsingular, show that the characteristic values of A^-1 are the reciprocals of A, and the A and A^-1 have the same characteristic vectors.

I need help getting started on this. Any help would be greatly appreciated. Thanks,

Jim

Fix an eigenvalue of A and let v be an eigenvector w.r.t. the eigenvalue, write down . Since A is invertible, is nonzero and you can multiply both sides of the equation by the inverse of A.