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Similar matrix proof involving idempotency

  1. Dec 12, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that if A is idempotent matrix of order n and B is similar to A, then B is idempotent






    3. The attempt at a solution

    I was just wondering if my attempt at the solution right here is correct.

    Given: A2=A and B=Q-1AQ

    Attempt: B=Q-1QA B=IA B=A

    Therefore, B=A^2 B=A(A) B=B(B) B=B2
     
  2. jcsd
  3. Dec 12, 2011 #2

    Dick

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    No, not right. Matrices don't necessarily commute. You can't just change AQ into QA. Just work out (Q^(-1)AQ)^2.
     
  4. Dec 12, 2011 #3

    Mark44

    Staff: Mentor

    The problem says that A is idempotent of order n, not 2.
    You can't do this (switch the order of A and Q). Matrix multiplication is generally not commutative.

    Start with B2 and show that it equals B.

    For the real problem, you'll probably need to use induction.
     
  5. Dec 12, 2011 #4
    If it says that B is similar to A, then am I allowed to assume A is similar to B a well?
     
  6. Dec 12, 2011 #5

    Mark44

    Staff: Mentor

    It's easy to show, but you don't need to do that. Dick and I have both suggested the same thing. Why don't you start with that?
     
  7. Dec 12, 2011 #6
    I know how to do it, but I'm just wondering if I can assume that if i'm only given that B is similar to A.
     
  8. Dec 12, 2011 #7

    Dick

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    Well, can you show that if B is similar to A then A is similar to B?
     
  9. Dec 12, 2011 #8
    B=q-1aq
    qb=q-1qaq
    q-1bq=q-1qaq-1q
    q-1bq=iia
    q-1bq=a

    the letters are not showing up in caps for some reason.
     
  10. Dec 12, 2011 #9

    Dick

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    I wouldn't worry about the caps. But you are juggling the order of the matrices around again to force things to work the way you want them to. They don't commute. Try that again without assuming the matrices commute.
     
  11. Dec 12, 2011 #10

    Mark44

    Staff: Mentor

    You can't do this (above). On the left side of the equation, you multiplied on the left by Q, but on the right side of the equation, you multiplied on the right. When you multiply both sides of an equation by a matrix, you have to multiply both sides in the same way - either both sides on the left or both sides on the right.
    You probably hit your CAPS LOCK key.
     
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