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Non commuting operators

  1. Sep 17, 2009 #1
    I have the following two equations

    #1

    d(A(t))/dt=A(t)B

    where A is some matrix that depends on parameter t, and B is another matrix, d is the differential

    this can be simplified to by multiplying both sides by the left inverse of A(t),

    A^-1(dA(t))=B*t

    which allows me to solve A(t) = Ce^(Bt)

    #2

    d(A(t))/dt=BA(t) note that A and B do not necessarily commute

    I'm asked to once again find A(t)

    and I get

    dA(t)A^-1(t)=Bdt

    but how do I integrate this thing?
     
  2. jcsd
  3. Sep 18, 2009 #2
    How about this?

    [itex]0 = \frac{d}{dt}I = \frac{d}{dt}\left(A(t)A^{-1}(t)\right) = \dot A(t)A^{-1}(t) + A(t)\dot{A^{-1}}(t) \ \ \forall t\in\mathbb{F}[/itex]

    Then,

    [itex] -A^{-1}(t)\dot A(t)A^{-1}(t) = \dot{A^{-1}(t)}[/itex]

    and

    [itex] \dot A(t) = -A\dot{A^{-1}}(t)A[/itex]
     
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