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Math Help - Please help

  1. #1
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    Please help

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  2. #2
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    Re: Please help

    Consider the matrices with a single 1 on the diagonal, 0 elsewhere. There are n of these. There are also \frac{n(n-1)}{2} matrices with two 1s in symmetric positions off diagonal and \frac{n(n-1)}{2} with a 1 and -1 in symmetric positions off diagonal (so the resultant matrix is antisymmetric).

    These form a basis for M_n(\mathbb{R}) (they are clearly linearly independent, and there are n^2 of them). The symmetric matrices we have defined has eigenvalue 1 under \varphi, and the antisymmetric matrices we have defined has eigenvalue -1. It should be obvious what the matrix of \varphi looks like with respect to this basis: n+\frac{n+1}{2} 1s and \frac{n(n-1)}{2} -1s down the diagonal, zero elsewhere.

    The determinant of \varphi is the product of eigenvalues: (-1)^{\frac{n(n-1)}{2}}
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