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Homework Help: Operator Example needed

  1. Mar 31, 2008 #1
    1. The problem statement, all variables and given/known data
    Give an example of an operator whose matrix with respect to some basis contains only 0's on the diagonal, but the operator is invertible.

    3. The attempt at a solution
    I think the operator will not have an upper-triangular matrix since it would then not be invertible.
     
  2. jcsd
  3. Mar 31, 2008 #2

    Dick

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    No, what you after is not upper triangular. What are some alternatives?
     
  4. Apr 1, 2008 #3
    i'm still unsure of what the diagonal values represent
    if the diagonals are all 0, what does that tell you about the operator?
     
  5. Apr 1, 2008 #4

    HallsofIvy

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    No body said all the diagonals are 0, just the main diagonal. Start simple. Try a 2 by 2 matrix with 0s on the diagonal. Now, you know that a matrix is invertible as long as its determinant is not 0. Just fill in the other two numbers so the determinant is not 0.
     
  6. Apr 1, 2008 #5
    our class was taught such that we're leaving determinants out until the end of the year...
    so that doesnt tell me a whole heck of a lot
    if the diagonal is all 0's, what condition would make the matrix invertible (aside from anything having to do with determinants) ?
     
  7. Apr 1, 2008 #6
    Well if it's a two dimensional matrix, A, you could easily find the inverse of it and show that [itex]AA^{-1}=I=A^{-1}A[/itex]. That way you wouldn't need to find its determinant.
     
  8. Apr 1, 2008 #7

    Dick

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    You must know SOME way to find the inverse of a matrix. Take M=[[0,a],[b,0]], put it into that method and figure out what conditions a and b need to satisfy for it to have an inverse.
     
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