# Frechet (second) derivative of the determinant and inverse functions

• Mathmos6
In summary, the conversation discusses the difficulties the speaker is facing while trying to evaluate limits for the Frechet derivatives of two functions: f(A)=A^-1 and g(A)=det(A). The speaker is specifically struggling with finding the Taylor series for f(I+H) around the identity matrix I and the second derivative of g(A) at I. They have tried using the chain rule and product rule, but have not been successful. They are seeking help and hoping to resolve these issues before Christmas.
Mathmos6
Hi all,
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)=A^-1 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, and I've evaluated the first derivative of f at A as Df(A)(H)=-A-1HA-1, 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 nth derivative in order to calculate the taylor series; 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!)

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)=Det(A)Tr(A-1H) (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 depressingly smart (/handsome ;)) people lend a hand?

Many thanks, I've spent hours on these two problems and they're getting to be quite an annoyance, so it'd be lovely to get them sorted before Christmas!

Last edited:
Anyone? :)

## 1. What is the Frechet (second) derivative of the determinant?

The Frechet (second) derivative of the determinant is a mathematical concept that describes the rate of change of the determinant of a matrix with respect to small changes in its entries. It is a second-order derivative, meaning it measures how the determinant's first-order derivatives change with respect to the matrix's entries.

## 2. Why is the Frechet (second) derivative of the determinant important?

The Frechet (second) derivative of the determinant is important because it allows us to analyze the sensitivity of the determinant to small changes in the matrix's entries. This is useful in many fields, including physics, engineering, and economics, where small changes in a system's parameters can have significant impacts on the system's behavior.

## 3. How is the Frechet (second) derivative of the determinant calculated?

The Frechet (second) derivative of the determinant can be calculated using standard matrix calculus techniques, such as the chain rule and product rule. It involves taking the derivative of the determinant's first-order derivative, which is known as the Jacobian matrix, and then taking the derivative of that result again.

## 4. What is the relationship between the Frechet (second) derivative of the determinant and the inverse function?

The Frechet (second) derivative of the determinant and the inverse function are related through the Inverse Function Theorem, which states that if a function is differentiable and its derivative is invertible at a point, then the function is locally invertible at that point with a unique inverse function whose derivative is the inverse of the original function's derivative. This theorem is useful for calculating the Frechet (second) derivative of the determinant for inverse functions.

## 5. Can the Frechet (second) derivative of the determinant be negative?

Yes, the Frechet (second) derivative of the determinant can be negative. The sign of the Frechet (second) derivative depends on the specific matrix and the values of its entries. A negative Frechet (second) derivative indicates that the determinant is decreasing with respect to changes in the matrix's entries, while a positive derivative indicates an increase. A zero derivative means that the determinant is not changing with respect to small changes in the entries.

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