Matrix Derivative Homework: Show Differentiability and Find Derivative

[Gavins]

Homework Statement

Let f: R^nxn -> R^nxn be the function f(A) = A^-1. ie f is the inverse function for some nxn matrix. Show it is differentiable and find the derivative.

Homework Equations

The Attempt at a Solution

I need to do something along the lines of (A + H)^-1 = A^-1 + f'(A)H + p(H) where p is a remainder term. I'm not sure how to expand this. I've looked in my notes and our lecturer did something along the lines of (A + H)^-1 - A^-1 = (A+H)^-1(I - (A+H)A^-1) = (A+H)^-1(-HA^-1). I can't really figure out what to do next since it doesn't seem easy to expand (A+H)^-1.

 

[fzero]

It might be easier to start with the relation f(A) A = I and differentiate both sides.

 

[Gavins]

f'(A)A + f(A) = 0?
f'(A) = -A^-2

That's assuming the derivative exists. Check if the remainder term goes to 0 to show it is differentiable. Let |.| be the norm.

(A + H)^-1 - A^-1 + A^-2H = p(H)
|p(H)|/|H| = |(A + H)^-1 - A^-1 + A^-2H|/|H|
=< |(A+H)^-1|/|H| - |A^-1|/|H| + |A^-2|

I don't think this is right since this definitely doesn't go to zero when H goes to zero. Sorry for the really bad format.

 

[fzero]

You can write f'(A) in terms of f(A). Determine the existence of f'(A) in terms of the existence of f(A).

 

[Gavins]

I'm not sure I understand. Is this easy to do from first principles? I think that's what my lecturer wants. Specifically, using the definition of the derivative, f(x+h) = f(x) + f'(x)h + p(h).

Thanks.

 

[D H]

You might want to step back and ask "What is the nature of this beast?" regarding the derivative of the inverse of some matrix A with respect to A.

Hint: It is not an n×n matrix.Given some invertible matrix A and a small perturbation δ of this matrix, assume that (A+δ) is invertible. Without loss of generality you can express the inverse of (A+δ) in the form (A^-1+ε). Derive an expression for this matrix ε.

Hint: Solve for (A^-1+ε)(A+δ)=I, assuming that the term εδ is small.What does this tell you about the desired derivative? What assumptions did you make that need to be validated?