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Check for projection operator

  1. Oct 10, 2014 #1
    If we are given an operator, say in matrix or outer product form, then how can we check if it is a projection operator? Is idempotence a sufficient condition for an operator to be a projection operator or are there any other conditions?
  2. jcsd
  3. Oct 10, 2014 #2


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    Gold Member

    O x O = O
  4. Oct 11, 2014 #3


    Staff: Mentor

    A positive operator P is a projection operator iff P=P^2.

    To see it note a projection operator has the form sum |bi><bi|. Square it and you get the same thing. Apply the spectral theorem to an operator P such that P=P^2 and we have sum pi |bi><bi| = sum pi^2 |bi><bi| which implies sum pi (1-pi) |bi><bi| = 0. Hence pi (1-pi) = 0 ie 1-pi = 0, pi =1.

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