Problems on left and right eigenvectors and eigenvalues

Question

Problem:

Complex Matrix, Diagonal Matrix, Left and Right Eigenvectors and Eigenvalues

1. Let X^-1 AX = D, where D is a diagonal matrix.

(a) Show that the columns of X are right eigenvectors and the conjugate rows of X^-1 are left eigenvectors of A.

(b) Let λ₁...,λ_n be the eigenvalues of A. Show that there are right eigenvectors x1,. . . , x and left eigenvectors y1, . . , yn such that

                              A = Σn ⁱ₌₁ λixiyi^*