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Thread: 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 $\displaystyle n$ of these. There are also $\displaystyle \frac{n(n-1)}{2}$ matrices with two 1s in symmetric positions off diagonal and $\displaystyle \frac{n(n-1)}{2}$ with a 1 and -1 in symmetric positions off diagonal (so the resultant matrix is antisymmetric).

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

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