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Describing Matrix/Transformation by Eigens.

  1. Mar 28, 2012 #1

    WWGD

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    Hi, All:

    Say T:R^n --->R^n is a linear map , and that the associated matrix M has a unique eigenvalue l=1. Is M necessarily a rotation matrix about the eigenspace?

    Thanks.
     
  2. jcsd
  3. Mar 28, 2012 #2

    morphism

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    No - e.g. consider ##\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right)##.
     
  4. Apr 2, 2012 #3

    WWGD

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    I see. I am trying to show that by applying kri+rj , i.e., adding a multiple of k times row i to

    row j one row to another row has the effect of rotating one of the k-planes about the

    solution subspace, since this is the only way I can conceive that the operation kri+rj

    preserves the solution to the system. Can you see how I else I can show this?
     
  5. Apr 4, 2012 #4

    morphism

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    Solution subspace of what?
     
  6. Apr 16, 2012 #5

    WWGD

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    Well, I have a soluble , i.e., non-contradictory (homogeneous) system of linear equations .

    The fundamental row operations--exchange rows, add a multiple of one

    row to another row-- preserve the solutions to the system. If we look at the

    solution S to the system geometrically, this is a subspace, possibly trivial. I'm trying

    to show that the operation of adding a multiple of row i to row j has the effect of

    rotating the n-planes in the system of equations about the solution- space S.

    I think the affine case--for non-homogeneous systems, is similar. I've been using

    the fundamental theorem of linear algebra that Bacle had mentioned in a similar

    problem, but I still can't prove this.
     
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