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Math Help - Linear Algebra II

  1. #1
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    Linear Algebra II

    Need some help.

    Find all eigenvalues and eigenvectors of the backward shift operator T Є L(F ∞) defined by:

    T(z1,z2,z3....) = (z2,z3....).


    Thanks
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  2. #2
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    Quote Originally Posted by taypez View Post
    Need some help.

    Find all eigenvalues and eigenvectors of the backward shift operator T Є L(F ∞) defined by:

    T(z1,z2,z3....) = (z2,z3....).


    Thanks
    Can you tell us in words what space is represented by L(F ∞)?

    I could guess, but would rather know.

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlank View Post
    Can you tell us in words what space is represented by L(F ∞)?

    I could guess, but would rather know.

    RonL
    Maybe all functions (not necesarrily continous) but defined on (-∞,∞)
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    Maybe all functions (not necesarrily continous) but defined on (-∞,∞)
    The L would normaly denote a space of linear operators, but I would
    expected a small "L" for one with a countable basis as I suspect this has.

    RonL
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  5. #5
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    Sorry. When I cut and pasted this into the post, it changed the script L.

    L(F) is the set of all linear maps on F infinity. F denotes R or C.

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  6. #6
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    Quote Originally Posted by taypez View Post
    Sorry. When I cut and pasted this into the post, it changed the script L.

    L(F) is the set of all linear maps on F infinity. F denotes R or C.

    Thanks
    Still not quite clear, my guess that we want L(F) here to be the space of
    linear operators on the space of all sequences on either R or C that are bounded?

    So if
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  7. #7
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    Quote Originally Posted by CaptainBlack View Post
    Still not quite clear, my guess that we want L(F) here to be the space of
    linear operators on the space of all sequences on either R or C that are bounded?

    So if
    If this is the case then T is the backwards shift operator which for all
    Z=(z(1), z(2), ..) in F with max(z(1), z(2), ..) = |Z| < infty:

    T(Z) = Z' = (z(2), z(3), .. )

    Now if lambda and X are eigen value and eigen vector for T, then:

    T(X) = lambda X,

    so x(n+1) = lambda x(n), for all n>1.

    Hence x(n) = lambda^(n-1) x(1), which is in F iff |lambda| <= 1.

    Thus every lambda such that |lambda|<=1 is an eigen value of T and
    (1, lambda, lambda^2, .., lambda^n, ..) is an eigen vector corresponding
    to the eigen value lambda.

    RonL
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