Derivative of the exponential map for matrices

  • Thread starter tolain
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  • #1
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Homework Statement


exp^\prime(0)B=B for all n by n matrices B.


Homework Equations


exp(A)= \sum_{k=0}^\infty A^k/k!


The Attempt at a Solution


Obviously I want to calculate the limit of some series, but I don't know what series to calculate. I wanted to try \lim_{h \to 0}(\exp(hB)-exp(0))/h, but I'm not sure if that's right. I feel like that's just taking the directional derivative, and not the entire derivative of exp. I'm just a little lost right now, and all would I like is not even solution but just help in pointing me in the right direction as to which limit I should be expliciting calculating.
 

Answers and Replies

  • #2
lanedance
Homework Helper
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First, what variable are you differentiating with respect to?

do you mean the derivative w.r.t. some variable, say t, of teh exponential matrix function
[tex] e^{At} [/tex]

or is A perhaps a function of some variable say t, A = A(t)?
 
  • #3
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I wanted to try \lim_{h \to 0}(\exp(hB)-exp(0))/h
That's good. But you can also write [itex]\exp (Bt)[/itex] as a power series an differentiate this power series term by term at t=0 (you would have to justify that differentiating term by term is ok here).

P.S. No. That's not good. Your notation suggest something else, a different meaning. You probably want to show that for any matrix A you have

[tex]\frac{d}{dt}\exp (B+tA)|_{t=0}=\exp(B).[/tex]

That is

[tex]\lim_{h\rightarrow 0}\left(\exp(B+hA)-\exp(B)\right)=\exp(B)[/tex]

The notation is not clear - see the previous comment by lanedance.
 
Last edited:
  • #4
lanedance
Homework Helper
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yeah so i think you want to look at
[tex] (\frac{d}{dt}e^{Bt})|_{t=0} [/tex]
 

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