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Show that Q is an idempotent matrix

  1. Nov 13, 2012 #1
    1. The problem statement, all variables and given/known data

    A square matrix P is called an idempotent if P^2 = P,

    Show that if P is an idempotent, so is Q = (P + AP - PAP) for any square matrix A (of the
    same size as P).

    2. Relevant equations



    3. The attempt at a solution

    basically I factor,

    Q = (I + (I - P) A) P

    then I square it and try to get back to the original,... and end up with

    Q^2 = (I + (I - P - P + P^2) A^2) P^2
    = (I + (I -2P + P) A^2) P
    = (I + (I - P) A^2) P
    = (I + AA - PAA) P
    = P + AAP - PAAP

    how do i get rid of the AA (A^2) ..?
     
    Last edited: Nov 13, 2012
  2. jcsd
  3. Nov 13, 2012 #2

    Mark44

    Staff: Mentor

    From your work below, you have apparently omitted a minus sign. The above should be:
    Q = (P + AP - PAP)

    Please confirm.
    I'm not sure that it's helpful to write Q in this form. Why not just multiply P + AP - PAP by itself?
     
    Last edited: Nov 13, 2012
  4. Nov 13, 2012 #3

    haruspex

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    Try that again, this time bearing in mind that matrix multiplication is not, in general, commutative.
     
  5. Nov 13, 2012 #4
    Uhhh... ah..

    Q^2 = ((I + (I - P) A) P) * ((I + (I - P) A) P)
    = Ok, I'm just going to leave it expanded. Too complicated this way

    = (P + AP - PAP)^2
    = (PP + PAP - PPAP + APP + APAP - APPAP - PAPP - PAPAP + PAPPAP)
    = (P + PAP - PAP + AP + APAP - APAP - PAP - PAPAP + PAPAP)
    = (P + AP - PAP)
    wow, that was easy. LOL.
     
  6. Nov 13, 2012 #5

    Mark44

    Staff: Mentor

    I rest my case.:approve:
     
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