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Matrix Symmetries

  1. Jul 24, 2005 #1
    I have a matrix A which satisfies: A_ij(+B) = A_ji (-B)
    (the matrix is a 4x4 matrix)
    EDIT: i also know that each row and line summes to 1.
    i want to prove that the inverse matrix of A sattisfies the same symmetry property. (But, with no success)

    Do you have an idea how to do that???

    thanks allot,
    Ron
     
    Last edited: Jul 24, 2005
  2. jcsd
  3. Jul 24, 2005 #2
    What is this (+B) and (-B) ?

    do you mean [tex]A_{ij} + B = A_{ji} - B[/tex] ?

    marlon
     
  4. Jul 24, 2005 #3
    sorry, i really didn't explain that...
    all matrix elements are functions of B...
    so - in the left side you enter +B, and in the right side, you enter -B
     
  5. Jul 24, 2005 #4

    matt grime

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    Let D be the inverse, what do we know?

    i will use summation convention, ie a repeated index is summed

    A_ij D_jk = d_ik (delta 1 if i=k 0 otherwise)


    now if i sum over i too it follows that the sum D_jk over j is 1, so that the sum of each column is 1 and similialy, by considering DA rather than AD it follows that the sum over each row of D is one.

    as for the other part, I know that

    A(b)D(b)=Id =D(b)A(b)

    so i can set b as -b if i feel like it and we still know

    D(-b)A(-b)=Id

    and i can transpose it

    A^t(-b)D^t(-b)=Id

    and i know that i can replace A^t(-b) with A(b), now i can reach the result i want.
     
  6. Jul 24, 2005 #5

    That's great!!! So ellegant...
    thanks allot!
     
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