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Derivative of a function whose variable is a matrix

  1. Feb 25, 2014 #1
    1. The problem statement, all variables and given/known data
    Let ##f: M_{n \times n} \rightarrow M_{n \times n}## with ##f(X) = X^2##, where ##M_{n \times n}## denotes the vector space of ##n \times n## matrices. Show ##f## is differentiable and find its differential.


    2. Relevant equations



    3. The attempt at a solution
    So far, I've been looking at the difference quotient in order to "guess" linear transformation ##A## that will satisfy it. We have:
    $$\lim_{|h|\to 0} \frac{|f(X+h) - f(X) - Ah|}{|h|} = \lim_{|h|\to 0} \frac{|Xh + hX + h^2 - Ah|}{|h|}$$after a little simplifying.
    And I was thinking for a fixed ##X##, I have ##A(h) = Xh + hX##, as I wanted to get rid of the ##Xh + hX## term in the quotient. However, I'm stuck now since it's ##Ah## and not just ##A##. I was thinking of throwing in an ##h^{-1}## to my ##A##, but that would make it non-linear. Any suggestions?
     
  2. jcsd
  3. Feb 25, 2014 #2

    Ray Vickson

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    The concept you need is that of the Frechet Derivative; see http://en.wikipedia.org/wiki/Fréchet_derivative or
    http://www.maths.lse.ac.uk/Courses/MA409/Notes-Part2.pdf . If ##h## is a matrix, then for ##f(x) = x^2## we have
    [tex] f(X+h) - f(X) = (X+h)(X+h) - X^2 = hX + Xh + h^2, [/tex]
    so the linear operator ##D_X: h \rightarrow M_{n \times n},## defined as ##D_X(h) = Xh + hX##, is the Frechet derivative. Since ##D_X## is linear, it will always be possible to write ##D_X(h)## as ##A_X h## for some matrix ##A_X##, if you really want to, but I am not sure if your question really requires that you do so.
     
  4. Feb 25, 2014 #3
    Ah okay, thank you! I just realized I was swapping linear transformation and matrix in my thought process, since in my linear algebra class "A" is normally a matrix while in my analysis class, "A" is a linear transformation. As such, it is saying A(h), not A*h. My confusion is cleared up =]
     
  5. Feb 26, 2014 #4

    Ray Vickson

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    I take back what I said about it always being possible to write ##D_X(h)## as ##A_Xh## for some matrix ##A_X##. I forgot that ##h## is a matrix, not a vector. If it were a vector, that re-write would be possible, but maybe not if it is a matrix. (Actually, ##h## is an ##n^2##-dimensional vector, so if we are willing to re-write all matrices as ##n^2##-vectors, there would, indeed, be an ##n^2 \times n^2## matrix giving us what we need, but that seems excessive.)
     
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