16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-27A(v) = A(αξ + βη)= αAξ + βAη= αr 1 ξ + βr 1 η= r 1 (αξ + βη)= r 1 vTherefore v is also an eigenvector as required.2. Let r 1 ≠ r 2 be distinct eigenvalues of A with associated eigenvectorsξ, η, respectively. Then, ξ is not a multiple of η.Proof.Assume that ξ = αη for some α. Since both vectors are not 0, we musthave α ≠ 0.Now,Aξ = r 1 ξ= r 1 αη,Aξ = Aαη = αAη = αr 2 η,So,r 1 αη = r 2 αη.Since α ≠ 0, and η ≠ 0, we get r 1 = r 2 which is a contradiction.3. A real matrix may not have any real eigenvalues, but always has complexeigenvalues.In the language of subspaces, if we consider all of the eigenvectors for r 1and add the zero vector, then we get a subspace of R n . This is called theeigenspace of r 1 .