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Operator in a real vector space has an upper block triangular matrix

  1. Mar 26, 2013 #1
    Hello All,

    I was trying to prove that an operator T in a real vector space V has an upper block triangular matrix with each block being 1 X 1 or 2 X 2 and without using induction.

    The procedure which i followed was :

    We already know that an operator in a real vector space has either a one dimensional invariant subspace or a 2 dimensional invariant subspace.

    Whatever be the case now, lets begin with the vector(s) which span these subspaces.

    Let U denote this subspace ----- (1)

    Now, if i am able to prove that there exists an another subspace W such that T is an invariant operator on the direct sum of U and W , then we can prove that operator T in a real vector space V has an upper block triangular matrix .

    I need a direction on proving the latter part.

    I sincerely thank you for the help.
     
    Last edited: Mar 26, 2013
  2. jcsd
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