Solving Difference Equations w/ Eigenvalues i & -i

[veronicak5678]

Homework Statement

The solution to du/dt = Au = [ 0 -1; 1 0] u (eigenvalues i and -i) goes around in a circle: u = (cos t, sin t). Suppose we approximate du/dt by forward, backward, and centered differences F, B, and C:
(F) U(n+1) - U(n) = AU(n)
(B) U(n+1) - U(n) = AU(n+1)
(C) U(n+1) - U(n) = .5A(U(n+1)+U(n))
Find the eigenvalues of I + A, (I-A)^-1, and (I-.5A)^-1(I+.5A). For which difference equation does the solution U(n) stay on a circle?

Homework Equations

Ax = lambda x

The Attempt at a Solution

I really don't know where to start...

 

[vela]

Take (F) for example. You can rewrite it as

U(n+1) = (I+A)U(n)

Note the matrix multiplying U(n) is the first matrix you're asked to find the eigenvalues for. You can do the same thing for the other two relations, and you should find that the matrices that multiply U(n) are the other two matrices.

What the problem wants you to do is start by finding the eigenvalues of those three matrices.

 

[veronicak5678]

Thanks for answering. For the eigenvalues I got:
i) 1+i & 1-i
ii) .5 + .5i & .5 - .5i
iii) 3/5 + 4/5i & 3/5 - 4/5i

Do they look OK? How do I know which stays on a circle?

 

[vela]

You found the correct eigenvalues. Say you found the corresponding eigenvectors as well. How would you find the solution for each equation?

 

[veronicak5678]

I) x1=[.707 -.707i] x2=[.707 .707i]
II) x1=[-.707i -.707] x2=[.707i -.707]
III)x1=[.707 -.707i] x2=[.707 .707i]

Do you mean I need to use the diagonal matrix and find the inverse of each of these?

 

[vela]

No, I mean if you consider these systems in the basis where the matrix is diagonal, you should easily be able to deduce how the solutions will evolve. It's not necessary to actually solve the systems this way, but that might be the easiest way to see what the answer is if you can't already see what the solutions will look like.