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Math Help - Center of a matrix group

  1. #1
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    Center of a matrix group

    The question is for me to find the center of a group comprising of non singular square matrices of order n.

    So if any member of the center is A, and G is any matrix in the group, we need AG = GA --> A = G-1 * A * G. (Where G-1 = G inverse)

    I can intuitively say that only elements of the form A = nI satisfy this equation, but how do I prove it?

    Please help
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  2. #2
    Super Member girdav's Avatar
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    Re: Center of a matrix group

    Let A a matrix in the center. First we will try to get information about the fact that AD=DA for all diagonal matrix D. We must have (AD)_{ij} = \alpha_ja_{ij} and (DA)_{ij} = a_{ij}\alpha_i if D = \mathrm{diag}(\alpha_1,\ldots,\alpha_n). If we fix i\neq j we must have a_{ij} =0 since we can choose a matrix D such that \alpha_i\neq \alpha_j. Hence A is a diagonal matrix: A =\mathrm{diag}(a_1,\ldots,a_n). Now if you take G with g_{ij} = i you will get that a_i=a_j for all i and j.
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  3. #3
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    Re: Center of a matrix group

    Any element in the center of the group must commute with an arbitrary matrix. The idea is to use very simple matrices, multiply on both sides, and get conditions that your matrix would have to satisfy so that it would commute with your simple matrix.

    These "simple matrices" will be elementary matrices; that is, matrices that correspond to elementary column (when multiplying on the right) or row (when multiplying from the left) operations. For example, if A is in the center of the group, then we should have

    \begin{pmatrix}-1&0&\cdots&\\0&1&0&\cdots\\0&0&1&\cdots\\\cdots&\c  dots&&\end{pmatrix}A=A\begin{pmatrix}-1&0&\cdots&\\0&1&0&\cdots\\0&0&1&\cdots\\\cdots&\c  dots&&\end{pmatrix}

    What does this tell us about A?

    (Note: That matrix is supposed to be I_n-2E_{1,1}, where I_n represents the n\times n identity matrix, and E_{i,j} the matrix with a 1 in position (i,j) and 0's elsewhere.)
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