Proving d(A^-1)/dL = -(A^-1)(dA/dL)(A^-1)

[chill_factor]

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?

 

[Ray Vickson]

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

 

[chill_factor]

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.