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Math Help - Find mod ( det A )

  1. #1
    MHF Contributor FernandoRevilla's Avatar
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    Find mod ( det A )

    Challenge problem

    Find \mod (\det A) being A\in\textrm{Mat}_{n\times n}(\mathbb{C}) defined by

    A=\begin{bmatrix} 1 & 1 & 1 & \ldots & 1\\ 1 &  \epsilon &  \epsilon^2 & \ldots &  \epsilon^{n-1} \\ 1 &  \epsilon^2 &  \epsilon^4 & \ldots &  \epsilon^{2n-2}\\ \vdots&&&&\vdots \\ 1 &  \epsilon^{n-1} &  \epsilon^{2(n-1)}&\ldots &  \epsilon^{(n-1)^2}\end{bmatrix}\quad (\epsilon=\cos(2\pi/n)+i\sin (2\pi/n))
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  2. #2
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    Re: Find mod ( det A )

    Quote Originally Posted by FernandoRevilla View Post
    Challenge problem

    Find \mod (\det A) being A\in\textrm{Mat}_{n\times n}(\mathbb{C}) defined by

    A=\begin{bmatrix} 1 & 1 & 1 & \ldots & 1\\ 1 &  \epsilon &  \epsilon^2 & \ldots &  \epsilon^{n-1} \\ 1 &  \epsilon^2 &  \epsilon^4 & \ldots &  \epsilon^{2n-2}\\ \vdots&&&&\vdots \\ 1 &  \epsilon^{n-1} &  \epsilon^{2(n-1)}&\ldots &  \epsilon^{(n-1)^2}\end{bmatrix}\quad (\epsilon=\cos(2\pi/n)+i\sin (2\pi/n))
    what does  \mod (\det A) mean? obviously \det A=\prod_{r < s}(\epsilon^s - \epsilon^r). maybe you want a more simplified answer?
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Re: Find mod ( det A )

    Quote Originally Posted by NonCommAlg View Post
    what does  \mod (\det A) mean?
    Modulus of the complex number \det (A) .

    obviously \det A=\prod_{r < s}(\epsilon^s - \epsilon^r).
    Right, A is a Vandermonde matrix.

    maybe you want a more simplified answer?
    Being this the Math Challenge Problem forum, I know the solution (and its resolution) . Another version is: prove that \mod (\det A)=n^{n/2} .
    Last edited by FernandoRevilla; December 18th 2011 at 01:06 AM.
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    Re: Find mod ( det A )

    Quote Originally Posted by FernandoRevilla View Post
    Modulus of the complex number \det (A) .

    Right, A is a Vandermonde matrix.

    Being this the Math Challenge Problem forum, I know the solution (and its resolution) . Another version is: prove that \mod (\det A)=n^{n/2} .
    so the question is to find |\det(A)|. so far i got  |\det(A)| =  2^{\binom{n}{2}}a^n, where a = \prod_{k=1}^m \sin(k \theta) and \theta = \frac{\pi}{n} and  m = \lfloor \frac{n-1}{2} \rfloor.

    so the problem now is to evaluate a.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Re: Find mod ( det A )

    A better approach is to find previously A^2 .
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