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The Spectral Theorem

  1. Feb 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that for any square matrix A, ker A= (im A*)[tex]\bot[/tex]

    2. Relevant equations

    A* = A conjugate transpose

    3. The attempt at a solution

    so we want to show that all A that satisfies Ax=0 for some x, that set is equal to (im A*)[tex]\bot[/tex]

    But i really dont know what image of this could possibly be... we arent given a function.
  2. jcsd
  3. Feb 1, 2009 #2
    it holds for any function.

    prove it for any function (you should be able to)
  4. Feb 1, 2009 #3
    So the image of the null space of A* is equal to the null space of A.

    Im thinking this proof leads me in the direction that im A* = A. then we are done. but i dont know how to show that.
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