1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Consider, first, the case in which A is diagonal. What do T and U do to the "basis" matrices?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook