Linear Algebra Proof 2

Question

You are told that a certain 2 x 2 matrix A has eigenvalues λ1 = 2 and λ2 = -1, with associated eigenvectors v1 = [1,1]^T , v2 = [1,2]^T .

(a) Find the matrix A.

(b) Find a closed formula for Aⁿ for any n.

Answer 1

Hi Kolten,

The simplest answer is to just take {v1,v2} as an ordered basis for R². With respect to this basis, A is simply

A = diag{2,-1} and A^n = diag{2ⁿ, (-1)ⁿ}. If P is the transition matrix from the ordered basis {v1,v2} to any other ordered basis, then with respect to this new basis A and Aⁿ are given by

A = P diag{2,-1} P^-1 and Aⁿ = P diag{2ⁿ, (-1)ⁿ} P^-1,

respectively.