The discussion centers on proving the derivative of the inverse of a matrix, specifically the equation d(A^-1)/dL = -(A^-1)(dA/dL)(A^-1). Participants explore the relationship between scalar functions and matrix derivatives, using the analogy of a function of numbers to derive the matrix form. The key approach involves defining B(x) as the inverse of A(x) and applying the product rule to the identity matrix. This leads to the conclusion that the derivative of the inverse matrix can be expressed in terms of the original matrix and its derivative.
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chill_factor said:Homework Statement
L = lambda.
Prove: d(A^-1)/dL = -(A^-1)(dA/dL)(A^-1)
Homework Equations
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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?
Ray Vickson said: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