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Primary Decomposition Theorem

  1. Feb 24, 2006 #1
    "Let [tex]m(x)[/tex] be the minimal polynomial of [tex]T:V\rightarrow V, \dim V<\infty[/tex] such that [tex]m(x)=m_1(x)m_2(x)[/tex] where [tex]gcd(m_1,m_2)=1[/tex], then there exists [tex]T[/tex]-invariant subspaces [tex]V_1, V_2[/tex] such that [tex]V=V_1\oplus V_2[/tex]."

    What other names is this thoerem called? It was given to me as the "primary decomposition theorem" but it's neither in my book nor in mathworld or wikipedia.
     
    Last edited: Feb 24, 2006
  2. jcsd
  3. Feb 24, 2006 #2

    mathwonk

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    look on my webpage, http://www.math.uga.edu/~roy/ and download my linear algebra notes. this is proved there.

    the proof is based on the linear combination proeprty for gcd's, i.e. you can write 1 as a linear combination of two polynbomilas with gcd = 1, so if 1 = pm1 + qm2, then plugging in T for X, we get id = p(T)m1q(T) +q(T)m2(T). so we have decomposed the identity map into two direct sum components. this decompsoes V accordingly. i.e. V1 = ker pm1, and V2 = ker qm2. is that right? (i am a little under the weather at the moment.)
     
  4. Feb 24, 2006 #3
    Yup, this is it. Thanks.
     
  5. Feb 24, 2006 #4

    mathwonk

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    my pleasure. we live in the hope of being of service, and occsionally this occurs.
     
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