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Math Help - Algebraic and Geometric multiplicity

  1. #1
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    Algebraic and Geometric multiplicity

    let A be the matrix
    |4 1 0 0 0|
    |0 4 1 0 0|
    |0 0 4 1 0|
    |0 0 0 4 1|
    |0 0 0 0 4|

    I am asked to find the algebraic multiplicity
    and the geometric multiplicity

    I think as it is an upper triangular matrix eigenvalues are the diagonal entries i.e. all 4,
    4^5 so Am is 5, im not sure on the geometric multiplicity though. Any help appreciated!
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  2. #2
    A Plied Mathematician
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    The geometric multiplicity is the dimension of the eigenspace corresponding to your eigenvalue. That is, when you solve

    (A-4 I)x=0,

    how many arbitrary parameters do you get? That'll be the geometric multiplicity.
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  3. #3
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    Ah ok, I've done that and I find x2=x3=x4=x5=0 leaving only arbitrary parameter x1 hence Gm is 1 and eigen-space is Sp(e1). Is that right?
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  4. #4
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    Sounds good to me. You can check your work here.
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