Compute an eigenvector for each of the eigenvalues found in

Question

1. (a) Compute all eigenvalues of the 3 x 3 matrix

(b) Compute an eigenvector for each of the eigenvalues found in part (a). Be sure to scale the eigenvectors so that each is a unit vector (i.e., v^T.v = 1 for each eigenvector v).

(c) Let the three (unit) eigenvectors be v_i, i = 1, 2, 3 with corresponding eigenvalues λ_i i = 1, 2, 3. Compute the following matrix

Note that the columns of the matrix [v₁, v₂, v₃] are the (unit) eigenvectors.

2. Suppose A is an N x N matrix with eigenvectors v_i, i = 1, 2, 3......
N and corresponding eigenvalues λ_i, i = 1, 2, 3........
N.

(a) Show that the characteristic polynomial det(A - λI) can be written as

det(A - λI) = (λ- λ₁) (λ-λ₂) (λ-λ₃)..........(λ-λ_N)

(b) Let M(λ) = det(A-λI). Prove that

(N - 1)!dN-1M(λ)/dλ^N-1|λ = 0 = (-1)^N-1(λ₁ + λ₂ + λ₃ +............ + λ_N)