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Eigenvectors and Eigenvalues watch

1. Okay, im having a few problems with Eigenvectors, any help would be appriciated

(ii) Suppose that x is an eigenvector of an invertible matrix A, with corresponding eigenvalue Show that x is also an eigenvector of (A + A^-1). What is the corresponding eigenvalue?

For this, can i put it into the equation Ax = I, but instead of A, put in A+A^-1?

(iii) Suppose that A is an n n matrix satisfying A^3 = -A. What are all possible eigenvalues of A?

For this, im not as sure, but could you make it A^3 +A, then simplify to A(A^2 +1)? im not too sure what to do from there.

(iv) Consider the generalised eigenvalue problem Ax = Bx, where A and B are n x n matrices. Show that the characteristic polynomial is det(A - B) = 0 in this case.

For this, i have just put B into the equation Ax = Ix, instead of I, then manipulated it.. but im not sure, it seems too simple?

Thanks for any help offered
2. (Original post by Jtt16)
(ii) Suppose that x is an eigenvector of an invertible matrix A, with corresponding eigenvalue Show that x is also an eigenvector of (A + A^-1). What is the corresponding eigenvalue?

For this, can i put it into the equation Ax = I, but instead of A, put in A+A^-1?

My linear algebra is way too rusty, so I'm only going to comment on (ii).

Don't know if this is what you meant:

You know

You are asked to show where you need to work out what the "?" is.

As a first step I'd work out what is.
3. For the second one, say Ax = kx (k scalar). A^3 = -A, so (A^3)x = -Ax.

Can you see where to go from there?
4. (Original post by Scipio90)
For the second one, say Ax = kx (k scalar). A^3 = -A, so (A^3)x = -Ax.

Can you see where to go from there?
take the -Ax over to the other side, then take x out?
5. (Original post by ghostwalker)
My linear algebra is way too rusty, so I'm only going to comment on (ii).

Don't know if this is what you meant:

You know

You are asked to show where you need to work out what the "?" is.

As a first step I'd work out what is.
so work out the eigenvalue for A^-1?
6. (Original post by Jtt16)
take the -Ax over to the other side, then take x out?
No, substitute in Ax = kx repeatedly to get an equation in k and x only.
7. (Original post by Scipio90)
No, substitute in Ax = kx repeatedly to get an equation in k and x only.
sorry i not quite sure what you mean..?
8. Using the left-hand side as an exxample:

(A^3)x = (A^2)(Ax) = ...

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