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Linear Algebra Proof

  1. Sep 6, 2012 #1
    1. The problem statement, all variables and given/known data

    L = lambda.

    Prove: d(A^-1)/dL = -(A^-1)(dA/dL)(A^-1)

    2. Relevant equations

    ???

    3. The attempt at a solution

    I did this as an analogy with function of numbers, but don't know how to extend this to matricies. for example:

    lets say A = f(L)

    d(f(L)^-1)/dL = - (f(L)^-2*d(f(L))/dL = -(A^-1)*dA/dL*(A^-1)

    But what is the matrix form?
     
  2. jcsd
  3. Sep 6, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I'll use x instead of L, and let B(x) = Inv(A(x)); thus, A(x)*B(x) = I (identity matrix). Take the derivative.

    RGV
     
  4. Sep 6, 2012 #3
    Thanks greatly.
     
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