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Projection matrices

  1. Feb 19, 2009 #1
    Out of the unit matrix and a real non-invertible symmetric matrix of the same size,

    [tex]\delta_{ij}[/tex] and [tex]M_{ij}[/tex]​

    I need to build a set of projection matrices, [itex]A_{ij}[/itex] and [itex]B_{ij}[/itex] which satisfy orthonormality:

    [tex]A_{ij} B_{jk}=0,[/tex] and [tex]A_{ij} A_{jk}=B_{ij} B_{jk}=\delta_{ik}[/tex]​

    Is this possible? or should I give up trying to find such matrices?
     
  2. jcsd
  3. Feb 22, 2009 #2
    Where does the matrix M come in?

    I don't think what you're requesting is possible. Just writing in terms of matrices, you want AB = 0 and A2 = B2 = I. But the first condition shows that det(A)det(B) = 0, so det(A) = 0 or det(B) = 0. If det(A) = 0, then det(A2) = 0, making A2 = I impossible.
     
  4. Feb 24, 2009 #3
    Good point; there are no such matrices I can construct. Thanks.
     
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