Matrix logarithm

1. Nov 25, 2013

hedipaldi

1. The problem statement, all variables and given/known data

Hi,
how can i show that the matrix logarithm log(I+A) is continuously differentiable on the set of matrices having operator norm less than 1.

2. Relevant equations

http://planetmath.org/matrixlogarithm

3. The attempt at a solution
i tried to compute the derivative but it is awkward

2. Nov 25, 2013

Ray Vickson

Computing the derivative may be hard, but all you are being asked to do is to show that the derivative exists and is continuous.

3. Nov 25, 2013

hedipaldi

And how do i do that?

4. Nov 25, 2013

Ray Vickson

I'm afraid I cannot help you there; I know how to do it, but it is not my homework. I would, however, suggest that you write out a few terms of the log and see if the resulting polynomial is continuously differentiable---again, without necessarily being able to compute it convenienetly.

5. Nov 25, 2013

hedipaldi

I mean,continuously differentiable as an operator,that is, the derivative is continuous as a function of the matrix A.I will be glad if you send me a more specific hint.

6. Nov 25, 2013

Ray Vickson

Define what is meant by the derivative in this case.