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Matrix Theory problem

  1. Mar 2, 2012 #1
    1. The problem statement, all variables and given/known data
    If B≈A with (P^-1)*A*P = B and also (Q^-1)*A*Q = B, show that Q=RP where R is a nonsingular matrix that commutes with A.


    2. Relevant equations
    AR = RA


    3. The attempt at a solution
    Our professor told us that it is easier to just play around with these equations and get the answer. That breaking them down to their elements would be a ton of work. I tried multiplying by a bunch of various ways. First I changed the equivalence formulas to, AP=PB and AQ=QB, and I tried multiplying by R, because of the commuting factor, but I couldn't get anywhere doing that. If someone could give me a hint as to if I'm missing a property or something that could help me out, that'd be great. Thanks.
     
  2. jcsd
  3. Mar 2, 2012 #2

    tiny-tim

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    hi arpitm08! :smile:

    hint: what happens if you multiply either equation by Q ? :wink:
     
  4. Mar 2, 2012 #3

    Deveno

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    i have an alternate approach:

    note that B = P-1AP is the same as: A = PBP-1.

    also, note that if R-1AR = A, then AR = RA.

    what happens if you evaluate (QP-1)-1A(QP-1)?
     
  5. Mar 2, 2012 #4

    tiny-tim

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    isn't that the same as mine, except using P instead of Q ? :wink:
     
  6. Mar 2, 2012 #5

    Deveno

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    your previous hint was:

    "multiply either equation by Q".

    i count 5 "=" in the OP's post, so it is unclear to me which two of them you mean. i suppose you mean:

    1) P-1AP = B
    2) Q-1AQ = B

    note that "multiply by Q" is not unambiguously defined, since Mat(n,F) is a non-commutative monoid. presumably you meant "left-multiply by Q".

    i think your IDEA is the same idea as mine, i think our hints are not.
     
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