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Math Help - Primary Decomposition theory

  1. #1
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    Primary Decomposition theory

    Dears,
    Could you please help me in proving the following:

    " Let T be a linear opearator on finite dimensional spsce V, let p = p_1^{{r_1}} \cdots p_k^{{r_k}} be a minimal polynomial for T, and let V = {W_1} \oplus  \cdots  \oplus {W_k} be the primary decomposition for T, i.e., W_i is the null space of
    {p_i}{(T)^{{r_i}}} . Let W be any subspace of V which is invarient under T. Prove that
    W = (W \cap {W_1}) \oplus (W \cap {W_2}) \ldots  \oplus (W \cap {W_k}) "

    With Best Wishes
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  2. #2
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    Re: Primary Decomposition theory

    The canonical decomposition of a vector space with respect to a linear transformation T can be found in any good linear algebra book. The solution to your problem is really just understanding this decomposition:

    Primary Decomposition theory-mhfvectorspace.png
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