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Idempotent Proof

  1. Sep 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that if (A^τ)A = A, then A is idempotent. [Hint: First show that (A^τ)A = A^τ]


    2. Relevant equations
    N/A


    3. The attempt at a solution
    I've gotten to the hint portion by taking the transpose of both sides, but have been unable to get that far past that. I've tried right side multiplying by A^-1 and have gotten this far:
    A^τ = A^τ(A^-1), then, taking the transpose of each side yields
    A = [(A^-1)^τ]A

    I can't figure out how to get rid of the transpose/inverse from there. Any help would be greatly appreciated. Thanks!
     
  2. jcsd
  3. Sep 13, 2012 #2

    jbunniii

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    You can't take the inverse because you don't know that A is invertible. Indeed, if A is invertible and idempotent, then A must be the identity matrix. (Can you prove that?)

    You are given that [itex]A^T A = A[/itex], and you have shown that [itex]A^T A = A^T[/itex]. The left hand sides of these two equations are the same, and therefore the right hand sides must also be the same. What does that imply about A?
     
  4. Sep 13, 2012 #3
    Ah, I see now. So A is equal to its own transpose so you can just substitute A back into the initial equation for A^T and get A^2 = A proving it's idempotent. I really appreciate the help!
     
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