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Inverse of the difference of two matrices A and B

  1. Oct 4, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that
    1/(A-B) = (1/A) + (1/A)B(1/A) + (1/A)B(1/A)B(1/A) + (1/A)B(1/A)B(1/A)B(1/A)+.....

    where A and B are matrices whose inverse exist.


    3. The attempt at a solution
    I tried to start from the LHS by pulling out (1/A):
    LHS = (1/A)[1 + B(1/A) + B(1/A)B(1/A) + B(1/A)B(1/A)B(1/A)+.....]
    = (1/A)*{1/[1-B(1/A)]} where I have used equation of GP sum to infinity
    = 1/(A - AB(1/A))
    However, i can't get the expression in RHS since A and B are not commute.

    Are there any other possible approach to this problem?

    Any help would be great~thanks~
     
    Last edited: Oct 5, 2008
  2. jcsd
  3. Oct 5, 2008 #2

    HallsofIvy

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    Re: Matrix

    So, more simply, (1/A)(1/1-(B/A))= (1/A)[(B/A)+ (B/A)2+ (B/A)3+ ...)

    Do you know the proof of the sum of a geometric series?>

     
  4. Oct 5, 2008 #3

    statdad

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    Re: Matrix

    The geometric series approach came to mind as well, but since these are matrices some care needs to be taken.
    If this is simply a formal problem the suggestion makes sense: usually, however, it is not enough to go through the mechanics - some conditions on the norms of the matrices are required.
    This seems to be a rather poorly worded problem
    (BUT, I agree with the suggestion of HallsofIvy - that seems to be the intent here)
     
  5. Oct 5, 2008 #4

    HallsofIvy

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    Re: Matrix

    You cannot assume that the formula for the sum of a geometric series of real numbers, but you can copy the proof- being careful that you don't assume commutativity.
     
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