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    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 \lambda 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 = \lambdaI, 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 = \lambdaBx, where A and B are n x n matrices. Show that the characteristic polynomial is det(A - \lambdaB) = 0 in this case.

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

    Thanks for any help offered
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    (Original post by Jtt16)
    (ii) Suppose that x is an eigenvector of an invertible matrix A, with corresponding eigenvalue \lambda 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 = \lambdaI, 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 Ax=\lambda x

    You are asked to show  (A+A^{-1})x= ?x where you need to work out what the "?" is.

    As a first step I'd work out what A^{-1}x is.
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    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?
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    (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?
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    (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 Ax=\lambda x

    You are asked to show  (A+A^{-1})x= ?x where you need to work out what the "?" is.

    As a first step I'd work out what A^{-1}x is.
    so work out the eigenvalue for A^-1?
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    (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.
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    (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..?
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    Using the left-hand side as an exxample:

    (A^3)x = (A^2)(Ax) = ...
 
 
 
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