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Math Help - Generators of matrix algebra

  1. #1
    Member Mauritzvdworm's Avatar
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    Generators of matrix algebra

    Let M_n(\mathbb{C}) be the space of all n\times n matrices with complex entries.

    Consider the matrices
    u=\left(\begin{array}{cccc}1    &0    &\cdots    &0\\0    &q    &\cdots    &\vdots\\0    &0    &\ddots    &\vdots\\0    &0    &\cdots    &q^{n-1}\end{array}\right) and v=\left(\begin{array}{ccccc}  0    &1    &0    &\cdots    &0\\0    &0    &1    &\cdots    &0\\0    &0    &\vdots    &\ddots    &0\\\vdots  &\vdots    &\cdots    &\cdots    &1\\1    &\cdots    &0    &\cdots    &0\end{array}\right)

    Where u^n=v^n=1 and q^n=1.

    Show that the matrices u \text{ and } v generate the whole of M_n(\mathbb{C}).

    The two by two case is simple, since you can easily find four linearly independant matrices which you can use as the basis. However, I am convinced there has to be some easier way? For the n by n case we would then need to find n^2 linearly independant elements which is a little cumbersome.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Re: Generators of matrix algebra

    You can certainly do it fairly efficiently if you are prepared to throw a bit of heavy machinery at it. Here is an outline.

    Step 1. The algebra A generated by u and v is selfadjoint, so by the double commutant theorem it will suffice to prove that the commutant of A consists only of scalars.

    Step 2. The subalgebra of A generated by u comprises all the diagonal matrices.

    Step 3. A matrix that commutes with all the diagonal matrices is itself diagonal. So the commutant of A consists of diagonal matrices.

    Step 4. A diagonal matrix that commutes with v must be a scalar.

    Edit. Step 2 assumes that q is a primitive n'th root of unity. If not, for example of q=1, then the result is false.
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  3. #3
    Member Mauritzvdworm's Avatar
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    Re: Generators of matrix algebra

    This is such a clever way to go about the problem!
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Generators of matrix algebra

    Quote Originally Posted by Mauritzvdworm View Post
    This is such a clever way to go about the problem!
    No wonder, Opalg is a clever man.
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