Derivative of the exponential map for matrices

[tolain]

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.

 

[lanedance]

First, what variable are you differentiating with respect to?

do you mean the derivative w.r.t. some variable, say t, of the exponential matrix function
$$e^{At}$$

or is A perhaps a function of some variable say t, A = A(t)?

 

[arkajad]

I wanted to try \lim_{h \to 0}(\exp(hB)-exp(0))/h

That's good. But you can also write $\exp (Bt)$ 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

$$\frac{d}{dt}\exp (B+tA)|_{t=0}=\exp(B).$$

That is

$$\lim_{h\rightarrow 0}\left(\exp(B+hA)-\exp(B)\right)=\exp(B)$$

The notation is not clear - see the previous comment by lanedance.

 

[lanedance]

yeah so i think you want to look at
$$(\frac{d}{dt}e^{Bt})|_{t=0}$$