I have the tridiagonal matrix (which comes from the backward Euler scheme)(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Finding eigenvalues

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