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Math Help - Eigen value of skew symmetric matrix

  1. #1
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    Eigen value of skew symmetric matrix

    We were trying to find the eigen value of a skew symmetric matrix:
    so we proceeded as :
    AX=kX X!=0 for some k as eigen value
    and A`=-A
    so A`+A = 0
    operating by X (matrix)
    A`X+AX = 0
    A`X+kX=0
    operating X`
    X`A`X+X`kX=0
    (AX)`X+X`kX = 0
    (kX)`X+X`kX=0

    here we ran into trouble as to the definition of k`.
    If we take k`=k
    we get that k=0 (i.e. eigen value is 0)
    But skew symmetric matrix can have 0 as well as imaginary eigen values, which we were unable to show.
    Is there some other way of doing it ?
    Thanks.
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  2. #2
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    Re: Eigen value of skew symmetric matrix

    Here k is just a constant so

    (kX)' = k X'

    So your expression gives

    2k|X|^2 = 0 and so k = 0

    Another thing you can do is to use

    A' = -A

    and just do

    A'X = -A X = -k X

    So the eigenvalues of A are either zero or they come in \pm pairs.
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