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Linear Algebra Proof

  • #1
901
3

Homework Statement



L = lambda.

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

Homework Equations



???

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?
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
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Homework Statement



L = lambda.

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

Homework Equations



???

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?
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
 
  • #3
901
3
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
Thanks greatly.
 

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