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Matrix diagonalization

  1. May 17, 2009 #1
    1. The problem statement, all variables and given/known data
    A =

    -10 6 3
    -26 16 8
    16 -10 -5

    B =

    0 -6 -16
    0 17 45
    0 -6 -16

    (a) Show that 0, -1 and 2 are eigenvalues both of A and of B .
    (b) Find invertible matrices P and Q so that (P^-1)*(A)*(P) = (Q^-1)*(B)*(Q)=

    0 0 0
    0 -1 0
    0 0 2

    (c) Find an invertible matrix R for which (R^-1)*(A)*(R) = B

    2. Relevant equations



    3. The attempt at a solution
    I was able to do Q1 and Q2 but not Q3.
    For Q2:
    P =
    0 1 1
    -1 2 3
    2 -1 -2

    Q =
    1 2 1
    0 -5 -3
    2 2 1

    Not really sure about Q3, since matrix B is not in the form I am used too.
    edit: I thought about it.
    using, (P^-1)*(A)*(P) = (Q^-1)*(B)*(Q)
    (Q)*(P^-1)*(A)*(P)*(Q^-1) = (Q)*(Q^-1)*(B)*(Q)*(Q^-1)
    (Q)*(P^-1)*(A)*(P)*(Q^-1) = (B)
    R = (P)*(Q^-1)
     
    Last edited: May 17, 2009
  2. jcsd
  3. May 17, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, of course! Now it's just a matter of finding Q^-1 and multiplying.
     
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