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Direct Sums

  1. Feb 11, 2006 #1
    "Suppose that T : V -> V is a linear transformation of vector spaces over
    R whose minimal polynomial has no multiple roots. Show that V can be
    expressed as a direct sum

    V = V1 + V2 + · · · + Vt

    of T-stable subspaces of dimensions at most 2. Show that, relative to a suitable basis, T can be represented by an n × n matrix with at most 2n non-zero entries, where n := dim(V)."

    Our professor is a little behind in lectures, but our assignments are still rolling full speed ahead. I'm not sure where to start. Can someone give me a hint?
     
    Last edited: Feb 11, 2006
  2. jcsd
  3. Feb 11, 2006 #2

    Hurkyl

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    Hrm. My first thought is the fundamental theorem of algebra.
     
  4. Feb 11, 2006 #3

    matt grime

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    If it were over C then of course you can diagonalize, but over R you can't. Remember roots will come in complex conjugate pairs because the minimal poly has real coefficients.

    Put that together....
     
  5. Feb 11, 2006 #4
    You're saying even if there are no real roots, I can somehow use a 2x2 matrix to "represent" a complex root in a real vector space? Like
    Code (Text):

     0 1
    -1 0
     
  6. Feb 12, 2006 #5

    matt grime

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    Not only am I saying that but so is the question.
     
  7. Feb 13, 2006 #6
    I can't get it right. What if the minimal poly has a single root only, where dim V>1?
     
  8. Feb 13, 2006 #7

    matt grime

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    If it is a single root then it is necessarily real, since the characteristic poly is over R. All roots occur as complex conjugate pairs. In this case the matirx is diagonalizable, over R.
     
  9. Feb 13, 2006 #8
    How can you conclude from the fact that it has a single root that it's diagonalizable over R?
     
  10. Feb 14, 2006 #9

    matt grime

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    Perhaps I was leaping to the conclusion that you meant single as in without multiplicity. If your 'minimal poly' of M has a single root, ie is m(x)=x-t, then M=tI. Since you have has a hypothesis that your minimal polynomial has no repeated roots we are in this situtation. Sorry if I didn't make it clear to you that I was using the hypotheses of the question. It is certainly not true that just because a matrix has one e-value it is diagonalizable, never mind over R or any other field. But we have the extra assumption here that we *have* to use.

    The minimal poly is real with no repeated roots, ie it is of the form

    m(x)=(x-a)(x-b)..(x-c)(x^2+dx+e)..(x^2+fx+g)

    where all numbers are real, no repeated roots and all quadratics have non-real roots.
     
  11. Feb 14, 2006 #10
    So by "not multiple" the question really means "non-repeating"? I've been doing it assuming that it was either 1 or no real roots.
     
  12. Feb 14, 2006 #11

    matt grime

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    Erm, just look up mutliple roots anywhere at all. I do not even know what you mean by 'one or no real roots'. Multiplicity of roots is a well defined concept if you ask me.
     
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