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Math Help - Matrices / Complex Numbers /eigenvalues+vectors

  1. #1
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    Exclamation Matrices / Complex Numbers /eigenvalues+vectors

    I really need some help answering this question, it's doing my head in!!

    Q:

    a) let z in the set of C (complex numbers). show that if z+zbar = 0, then Re(z) = 0 (something to do with Argand diagrams?)

    (where zbar is z with a horizontal line across the top :S)

    b) let A in the set of R^nxn (a square matrix, n by n) be skew-symmetric, that is Atransposed = -A. show that all eigenvalues of A have zero real-part.

    c) suppose that x and y are eigenvectors corresponding to distinct eigenvalues of a real-values skew-symmetric matrix. show that (xbartransposed)y=0


    can anyone help me solve this please?
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  2. #2
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    Quote Originally Posted by thatgirlrocks View Post
    I really need some help answering this question, it's doing my head in!!

    Q:

    a) let z in the set of C (complex numbers). show that if z+zbar = 0, then Re(z) = 0 (something to do with Argand diagrams?)

    (where zbar is z with a horizontal line across the top :S)

    [snip]
    Let z = x + iy and solve for x.

    Quote Originally Posted by thatgirlrocks View Post
    [snip]
    b) let A in the set of R^nxn (a square matrix, n by n) be skew-symmetric, that is Atransposed = -A. show that all eigenvalues of A have zero real-part.
    [snip]
    Apply a well known theorem about the eigenvalues of A and A^T.

    Quote Originally Posted by thatgirlrocks View Post
    [snip]
    c) suppose that x and y are eigenvectors corresponding to distinct eigenvalues of a real-values skew-symmetric matrix. show that (xbartransposed)y=0


    can anyone help me solve this please?
    Read this: http://www.math.umn.edu/~garrett/m/algebra/notes/24.pdf
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