I have the tridiagonal matrix (which comes from the backward Euler scheme) A = [ 1+2M - M 0 .... ] [ -M 1+2M 0 .... ] [ ..... ] [ -M 1+2M ] I am given that the eigenvectors are v_k = [ sin(pi*k*x_j ] for j = 1:J-1 and k = 1:J-1 I am to find the eigenvalues L_k. I tried just writing A * v_k = L_k * v_k Then doing the multiplication on the left hand side I can get a relationship from one equation: -M*sin(sin(pi*k*x_j-1) + (1+2M)sin(pi*k*x_j) -M*sin(pi*k*x_j+1) = L_k * sin(pi*k*x_j) I get stuck here. Is this the right approach? Is there someway to solve for L_k? Also: I am to show the v_k are orthogonal. Would I use dot product or the integral inner product to show this? Thanks for any help.