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Thread: Generators of matrix algebra

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

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

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

    Where $\displaystyle u^n=v^n=1$ and $\displaystyle q^n=1$.

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

    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 $\displaystyle n^2$ 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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