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I Proving Matrix exponential property

  1. Oct 8, 2016 #1
    this is not a homework question, I just want to make sense of the equation here.
    Assuming matrix A is diagonal,

    If A_hat=T'AT where T' is an inverse matrix of T.

    e^(A_hat*t)=T'e^(At)T
    which implies,
    e^(T'AT*t)=T'e^(At)T

    we know that e^(At) is a linear mapping, therefore if we convert f to some linear transformation P,
    PT'AT=T'PAT (not sure if this step is correct) this condition should be always true, but why?


    can anyone provide me a short proof of this?
     

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  2. jcsd
  3. Oct 11, 2016 #2
    The Taylor expansion of e^x is:
    [tex] e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... [/tex]
    So when x is a matrix T'AT, you get a lot of cancellations
    For example in the x^4/4! term, you get
    [tex] (T'AT)^4/4! = T'AT T'AT T'AT T'AT /4! = T' A^4 T /4! [/tex]
    This same kind of TT' = 1 cancellation happens in every term until you are left with T' on the left and T on the right of every term.
    So you get:
    [tex] T' (1 + A + A^2/2! + A^3/3! + A^4/4! + ...) T = T'e^A T [/tex]
     
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