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Thread: Generalized invariant subspace

  1. #1
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    Generalized invariant subspace

    Hi,
    Let A be an nxn non-scalar matrix (i.e A is not a scalar multiple of the identity matrix).Show that there exists a non-trinial subspace W of R^n ,

    that is invariant under every nxn matrix B such that AB=BA.

    https://en.wikipedia.org/wiki/Invariant_subspace
    Thank's in advance
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  2. #2
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    Re: Generalized invariant subspace

    Hey hedi.

    This should preserve information in a way that is one to one.

    Hint - Consider the determinant and how that helps figure out whether the transformation is one to one.
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  3. #3
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    Re: Generalized invariant subspace

    I dont see the connection to "one to one"
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  4. #4
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    Re: Generalized invariant subspace

    If v e W and T(v) e W then the information is preserved and the mapping has to be one to one.

    Since it goes from R^n to R^n you have a square matrix and because of that it has to have a non-zero determinant.
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  5. #5
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    Re: Generalized invariant subspace

    It has been a very long time that I have done this problem. I seem to recall showing that there must exist some positive integer $K$ such that for all $k\ge K$, we have $\text{dim}\left(A^k\mathbb{R}^n\right)=\text{dim} \left(A^K\mathbb{R}^n\right)$

    I completely forget how we used that to prove the theorem, though. I'll take another look at it after work.
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