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Conjugate transpose matrix

  1. Jan 6, 2008 #1
    Let A be a nxn matrix. Prove that if (A*)A=0 then A=0. What if A(A*) = 0?

    A* is the conjugate transpose of A. When I write out the expansion formula, I cannot conclude that every entry of A is zero. What am I missing?
     
  2. jcsd
  3. Jan 6, 2008 #2

    Dick

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    Concentrate on the diagonal elements of (A*)A. Each one is the inner product of a row of A* with the corresponding column of A. I.e. it's the inner product of a vector with it's conjugate transpose. Under what conditions can that be zero?
     
  4. Jan 6, 2008 #3
    Oh, that was a clever idea. I got it now. So it turns out the weaker condition of (A*)A having zero diagaonal entries is enough to conclude that A=0. And the same is true of A(A*).
     
  5. Jan 6, 2008 #4

    Dick

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    Bingo. You've got it.
     
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