Let A M2E be a 2 times 2 matrix with one single eigenvalue l

Let A M_2(E) be a 2 times 2 matrix with one single eigenvalue lambda and one single eigenvector v. We denote by w the generalized vector such that (Lambda - lambda I) w = v. Prove that v and w are linearly independent.

Solution

Let w and v be linearly dependent. Then there exist non-zero scalars a and b such that av+bw = 0. Then v =( -b/a)v. Further, since is an eigenvalue of A, we have (A- I)v = 0 or, (A- I)(-b/a)w = 0 or, (-b/a) (A- I)w = 0 so that (A- I)w = 0. However, this is a contradiction as (A- I)w =v. Hence v and w are linearly independent.