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Show that the range of the 2 matrices are the same

  1. Feb 20, 2016 #1
    1. The problem statement, all variables and given/known data
    ##P=A(A^*A)^{-1}A^*##
    where A is a mxn real/complex matrix and ##A^*A## is invertible.
    ##A^*## means the conjugate transpose of A.

    2. Relevant equations


    3. The attempt at a solution
    Let y be in the range(A), such that
    ##y = Ax## for some ##x##.
    We can see that ##PA = A(A^*A)^{-1}A^*A = A##
    Then
    ##y = PAx = P(Ax)##
    Does this expression above alone show that y is also in the range(P)?
     
  2. jcsd
  3. Feb 20, 2016 #2

    andrewkirk

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    That last expression is not correct.
    It should be ##Py = PAx = P(Ax)##.
     
  4. Feb 20, 2016 #3
    But previous I had shown ##PA=A##
    so ##y=Ax=PAx=P(Ax)##
    Why is that not correct?
     
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