Real analysis differentiation of a real function defined by a matrix

  1. 1. The problem statement, all variables and given/known data
    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


    2. Relevant equations



    3. 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!
     
  2. jcsd
  3. Hurkyl

    Hurkyl 16,090
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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?
     
  4. 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...
     
  5. Hurkyl

    Hurkyl 16,090
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    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: Dec 8, 2008
  6. 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...
     
  7. Hurkyl

    Hurkyl 16,090
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    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: Dec 8, 2008
  8. Hurkyl

    Hurkyl 16,090
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    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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