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Math Help - Matrices represented by Symmetric/Skew Symmetric Matrices

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    Matrices represented by Symmetric/Skew Symmetric Matrices

    I need to show that ever square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix. I'm completely confused on this question... Should I answer this by filling in an actual matrix with variables, and then somehow come up with the appropriate symmetric matrices, or what?
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  2. #2
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    Quote Originally Posted by Hellreaver View Post
    I need to show that ever square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix. I'm completely confused on this question... Should I answer this by filling in an actual matrix with variables, and then somehow come up with the appropriate symmetric matrices, or what?
    It will help you (very much) to write: A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T), where A^T is the transpose matrix of A.
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    But what do I need to do with that? I'm so confused... Sorry for my ignorance on the subject... I just don't get some of these things at all...
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    Quote Originally Posted by Hellreaver View Post
    But what do I need to do with that? I'm so confused..
    You asked to write a matrix as a sum of a symettric matrix and a skew symettric matrix.
    That is what Laurent did. Just check that the way he wrote it is what you want.
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    Would I actually need to write the matrix out, though, or is it ok to leave it like that?
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    Quote Originally Posted by Hellreaver View Post
    Would I actually need to write the matrix out, though, or is it ok to leave it like that?
    Just write A = B + C where B = \tfrac{1}{2}(A+A^T) (symmetric) and C = \tfrac{1}{2}(A-A^T) (skew symmetric).
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  7. #7
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    Try starting with A=\frac{1}{2}(A+A^{T})+\frac{1}{2}(A-A^{T})

    So, we only have to prove that \frac{1}{2}(A+A^{T}) is

    symmetric and that \frac{1}{2}(A-A^{T}) is skew symmteric.

    Note that:

    \frac{1}{2}(A+A^{T})^{T}=\frac{1}{2}(A^{T}+(A^{T})  ^{T})=\frac{1}{2}(A+A^{T})

    Now, you do it for the other case, \frac{1}{2}(A-A^{T}), and you're pretty much done.
    Last edited by galactus; October 25th 2008 at 06:00 PM. Reason: Beat to the punch while typing. Oops.
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  8. #8
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    Well, the big reason I was asking is because I will need to be able to solve instances with actual values:

    1 0
    2 1

    I need to represent this matrix as a sum of a symmetric matrix and a skew symmetric matrix.
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