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Prove that a linear operator is indecomposable

  1. Dec 2, 2013 #1
    1. The problem statement, all variables and given/known data

    Let V be a fi nite-dimensional vector space over F, and let T : V -> V be a linear operator. Prove that T is indecomposable if and only if there is a unique maximal T-invariant proper subspace of V.

    2. Relevant equations



    3. The attempt at a solution
    I tried using the definition of decomposable with respect to matrices, but I can't manipulate it to answer this question.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 2, 2013 #2
    Start by providing us the definition of "indecomposable operator" and "T-maximal".
     
  4. Dec 2, 2013 #3
    A n×n matrix A is decomposable if there exists a nonempty proper subset I⊆{1,2,...,n} such that aij=0 whenever i∈I and j∉I.

    I only know the definition of maximal vector which is: A vector z such that the minimal polynomial of the operator T with respect to z = the minimal polynomial of the operator T, is called a maximal vector.
     
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