Real analysis differentiation of a real function defined by a matrix

In summary, the student is trying to solve a homework problem, but is lost. They know the basics of derivatives, and that if a function is differentiable at a point, there exists a linear map and a remainder function. They are not sure why vTAv is confusing them, and are looking for help.
  • #1
Numbnut247
26
0

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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  • #2
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?
 
  • #3
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...
 
  • #4
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?
 
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  • #5
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...
 
  • #6
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"?)
 
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  • #7
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?​
 

1. What is real analysis differentiation?

Real analysis differentiation is a mathematical concept that involves calculating the rate of change of a real function. It is used to determine the slope of a curve at a specific point, and can also be used to find maximum and minimum values of a function.

2. How is differentiation of a real function defined by a matrix different from regular differentiation?

Differentiation of a real function defined by a matrix involves using matrix operations and properties to calculate the derivative of the function. This is different from regular differentiation, which only involves using algebraic operations.

3. What is the purpose of using a matrix in defining a real function for differentiation?

Using a matrix in defining a real function allows for a more general and versatile approach to differentiation. It allows for the differentiation of functions with multiple variables and can also handle more complex functions that cannot be easily differentiated using traditional methods.

4. Can real analysis differentiation be applied in real-world scenarios?

Yes, real analysis differentiation has many practical applications in fields such as physics, engineering, economics, and statistics. It can be used to model and analyze real-world phenomena and make predictions based on the rate of change of a function.

5. What are some common techniques used in real analysis differentiation of functions defined by a matrix?

Some common techniques used in real analysis differentiation of functions defined by a matrix include the chain rule, product rule, and quotient rule. Other techniques, such as implicit differentiation and partial derivatives, may also be used depending on the complexity of the function.

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