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Homework Help: Diagonalizable proof

  1. Nov 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Let V be the space of n X n matrices over F. Let A be a fixed n X n matrix
    over F. Let T and U be the linear operators on V defined by
    T(B) = AB
    U(B) = AB - BA.
    1. True or false? If A is diagonalizable (over F), then T is diagonalizable.
    2. True or false? If A is diagonalizable, then U is diagonalizable
    Thanks for the help.

    3. The attempt at a solution
    I'm guessing that 1 is true and 2 is false. I'm not sure, since these are linear operators rather than simple matrices.
  2. jcsd
  3. Nov 26, 2009 #2


    User Avatar
    Science Advisor

    Consider, first, the case in which A is diagonal. What do T and U do to the "basis" matrices?
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