Matrix Logarithm: Proving Continuity for Operator Norm < 1

[hedipaldi]

Homework Statement

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.

Homework Equations

http://planetmath.org/matrixlogarithm

The Attempt at a Solution

i tried to compute the derivative but it is awkward

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

http://planetmath.org/matrixlogarithm

The Attempt at a Solution

i tried to compute the derivative but it is awkward

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

 

[hedipaldi]

And how do i do that?

 

[Ray Vickson]

And how do i do that?

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.

 

[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.

 

[Ray Vickson]

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.

Define what is meant by the derivative in this case.