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Math Help - Eigenvectors and Values for T(A)=A^t

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    Eigenvectors and Values for T(A)=A^t

    I'm having some trouble getting started with this, if someone could point me in the right direction that would awesome.
    If T(A) = A^t (A-transpose) show that +1 and - 1 are the only eigenvalues of T.
    I think that my problem is that I don't know how to symbolize the matrix of linear transformation for T, but perhaps there is another way to go about doing this problem.
    Thanks!
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    Accidental post
    Last edited by HallsofIvy; May 14th 2010 at 01:29 AM.
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    T is a linear operation on matrices that maps a matrix into its transpose? I started to say "Think about the fact that A and A^T have the same main diagonal", but then it occured to me that A might not be a square matrix. Certainly, you can look at a_{11} and a_22 which must be the same in both matrices.
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    Quote Originally Posted by kaelbu View Post
    I'm having some trouble getting started with this, if someone could point me in the right direction that would awesome.
    If T(A) = A^t (A-transpose) show that +1 and - 1 are the only eigenvalues of T.
    I think that my problem is that I don't know how to symbolize the matrix of linear transformation for T, but perhaps there is another way to go about doing this problem.
    Hint: T^2 is the identity.
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    Quote Originally Posted by HallsofIvy View Post
    I started to say "Think about the fact that A and A^T have the same main diagonal", but then it occured to me that A might not be a square matrix.
    Isn't it true that for a matrix to have transpose it must be square?
    Also, I the fact that A is a square matrix was actually part of the problem (T is a linear operator on M_(nxn)(F)).
    Sorry I did not include that in the original problem, I guess I have a bad habit of over looking that part of the problem because I don't really have any idea what's going on, therefore it doesn't really make any difference to me what vector space the transformations in.
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    Oh!
    If T^2 is the identity, T must also be the identity, I think. Oh no, each diagonal entry must be the \sqrt(1) = + or - 1 .
    Thanks! You're the greatest!
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    Quote Originally Posted by kaelbu View Post
    Oh!
    If T^2 is the identity, T must also be the identity, I think. Oh no, each diagonal entry must be the \sqrt(1) = + or - 1 .
    Thanks! You're the greatest!
    Thanks for the compliment, but I'm not sure what you mean by the diagonal entries of T?

    T is defined by T(A) = A^{\textsc{t}}. So T^2(A)= (A^{\textsc{t}})^{\textsc{t}} = A. And \lambda is an eigenvalue of T if there is a nonzero matrix A such that T(A) = \lambda A. Then A = T^2(A) = T(T(A)) = T(\lambda A) = \lambda T(A) = \lambda^2A. So (\lambda^2-1)A = 0 and hence \lambda^2=1.
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    That's more or less what I meant, though I did it slightly differently.
    Thanks again for the help.
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