Implicit Function Theorem Tricky Proof on matrices?

[aeronautical]

Implicit Function Theorem ...Tricky Proof on matrices!?

Homework Statement

Implicit Function Theorem ... Tricky Proof on matrices!?
Show with the Implicit Function Theorem, that an n x n matrix B, can be solved for as a continuous differentiable function of a matrix A (which is n x n), from the equation BAB = id, given that || A - id || is sufficiently small.

Can someone outline the solution to this problem? I don't know how to proceed with the proof.

Can you please show all the steps of the proof?

 

[lippyka]

Homework Equations BAB = id|| A - id || is sufficiently smallThe Attempt at a Solution Let B(A) be a continuous differentiable function of A.We want to show that B can be solved for from the equation BAB = id when || A - id || is sufficiently small. We will use the Implicit Function Theorem for this. The Implicit Function Theorem states that if we have an equation of the form F(X,Y) = 0 and F is differentiable with respect to Y then there is an implicit function Y = f(X) such that F(X,Y) = 0. In our case, we have an equation BAB = id. Let F(A,B) = BAB - id. Then F is differentiable with respect to B. By the Implicit Function Theorem, there is an implicit function B = f(A) such that BAB = id. Now since || A - id || is sufficiently small, we can conclude that B = f(A) is a continuous differentiable function of A and that B can be solved for from the equation BAB = id when || A - id || is sufficiently small.