Real analysis differentiation of a real function defined by a matrix

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Homework Help Overview

The problem involves differentiating a real-valued function defined by a quadratic form associated with a real nxn matrix. The function is expressed as f(v) = v^tAv, where v is a vector in R^n. The task is to prove a specific expression for the derivative of this function.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the differentiation rules and the concept of transposes in the context of matrix calculus. Some express confusion about applying differentiation rules to the function and the implications of the transpose. Others question the foundational knowledge required to approach the problem, particularly regarding the definition of derivatives for vector functions.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts on the necessary differentiation concepts and the relationship between the function and its derivative. Some guidance has been provided regarding the relevance of known differentiation rules, but there is no clear consensus on how to proceed with the problem.

Contextual Notes

Participants note that they have not yet proven many differentiation rules, which may limit their ability to tackle the problem effectively. There is also mention of the expectation to verify the derivative rather than derive it independently.

Numbnut247
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Homework Statement


Suppose A is a real nxn matrix and f: R^n --> R is definted by f(v)=v^tAv (where v^t denotes the transpose of v). Prove that the derivative of f satisfies

(f'(v))(w) = v^t (A+A^t)w


Homework Equations





The Attempt at a Solution


I'm kinda lost here and I really don't know where to start. I know I have to show that the derivative "is" the linear map v^t(A+A^t) but I think the transpose is confusing me. Thanks in advance!
 
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The key things to remember are
. The differentiation rules
. Every 1x1 matrix is its own transpose

I'm not sure why you didn't think of simply trying to apply the differentiation rules to vTAv. Isn't that normally the first thing you think of for a differentiation problem?
 
Hurkyl said:
The key things to remember are
. The differentiation rules
. Every 1x1 matrix is its own transpose

I'm not sure why you didn't think of simply trying to apply the differentiation rules to vTAv. Isn't that normally the first thing you think of for a differentiation problem?
uh... we never proved any differentiation rules yet:S but i think you are referring to the product rule? but i don't know how they work in R^n or with linear maps. I'm really lost actually... haha. I don't get how i can somehow use the 1x1 matrix thing, either...
 
Well, if you haven't really proven much about derivatives, and you're expected to solve this problem... that means the few things you do know should be enough!

So what do you know about derivatives of vector functions? The definition, at least?
 
Last edited:
Hurkyl said:
Well, if you haven't really proven much about derivatives, and you're expected to solve this problem... that means the few things you do know should be enough!

So what do you know about derivatives of vector functions? The definition, at least?
i know if f is differentiable at a point x, there exists a linear map and a remainder function r which is continuous at 0 and r(0)=0. i know if f is linear, then it's multiplication by a matrix and the matrix is the derivative of f but there's the v transpose which confuses me...
 
I bet you also know an explicit formula relating the function, the derivative, and the remainder.

(p.s. is that an "if" or an "if and only if"?)
 
Last edited:
p.p.s. just to make sure it's clear, since a lot of people overlook it -- the problem you are asked to answer is
Verify that this function is the derivative of that function.​
You were not asked to answer
How would you have figured out that this function is the derivative of that function if you weren't told what it is?​
 

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