I'm trying to get to grips with the Frechet derivative, and whilst I think I've got all the fundamental concepts down, I'm having trouble evaluating some of the trickier limits I've come up against.
The two I'm struggling with currently are the further derivatives of the functions f(A)= on invertible matrices and g(A)=det(A) on all matrices.
For the former, I'm trying to find the Taylor series of f(I+H) about the identity matrix I, so I need to find the general form for a derivative and I've evaluated the first derivative of f at A as Df(A)(H)= , by using the chain rule in composition with 2 other functions j(A)=A(B^-1) and =(B^-1)A, but I'm having trouble evaluating further derivatives: I've spent a lot of time looking at Df(A+K)(H)-Df(A)(H), but to no avail, and not only that but I need to find a general formula for the n-th derivative in order to calculate the taylor series (unless at some point they become 0, but that seems unlikely!); can anyone suggest how I could get my hands on a general formula? (I'd use induction but I have no idea what I'd be hypothesizing, and if i did use it it would be nice to have a reason why I should think that's the right thing to hypothesize!)
For the latter, I want to find the second derivative of the determinant function at I (just the second derivative this time, not all of them!); I've calculated the first derivative at A to be Dg(A)(H)= (or just = Tr(H), at I) but once again I can't work out a nice way (or indeed, any way) to evaluate the k->0 limit of Dg(A+K)(H)-Dg(A)(H) and find the second derivative (could I use the product rule on Det and Tr separately? In that case, I could use a hand calculating the derivative of the trace, since I tried that too already!): could any of you exceedingly smart and handsome () people lend a hand?
Many many thanks, I've spent many hours on these two problems and they're getting to be quite an annoyance, so it'd be lovely to get them sorted before the New Year!