Let A = [ jk ] be an n x n matrix with (not necessarily distinct) eigenvalues

Let A = [α_jk] be an n x n matrix with (not necessarily distinct) eigenvalues λ₁, · · · , λ_n. Show.

kA has the eigenvalues kλ₁,· · ·, kλ_n. A^m(m = 1, 2, · · ·) has the eigenvalues λ₁^m, · · ·,λ_n^m. The eigenvectors are those of A.