Matrix Logarithm: Proving Continuity for Operator Norm < 1

In summary: Then, show that the resulting function is continuous on the set of matrices with operator norm less than 1.In summary, the conversation is about showing that the matrix logarithm log(I+A) is continuously differentiable on the set of matrices with operator norm less than 1. The student has attempted to compute the derivative but found it difficult and is now asking for help. The suggested approach is to write out a few terms of the log and show that the resulting polynomial is continuously differentiable. However, the student requests a more specific hint on how to define and show the derivative is continuous in this case.
  • #1
hedipaldi
210
0

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
 
Physics news on Phys.org
  • #2
hedipaldi said:

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.
 
  • #3
And how do i do that?
 
  • #4
hedipaldi said:
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.
 
  • #5
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
hedipaldi said:
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.
 

FAQ: Matrix Logarithm: Proving Continuity for Operator Norm < 1

What is the definition of matrix logarithm?

The matrix logarithm of a square matrix A is defined as the matrix B such that e^B = A, where e is the base of the natural logarithm.

What is the significance of proving continuity for operator norm < 1?

Proving continuity for operator norm < 1 is important because it ensures that the matrix logarithm function is well-defined for a wide range of matrices. It also allows for the use of analytical methods to solve equations involving matrix logarithms.

How is the continuity of matrix logarithm for operator norm < 1 proven?

The continuity of matrix logarithm for operator norm < 1 is proven using the Cauchy integral formula and the properties of analytic functions.

What is the importance of the operator norm in matrix logarithm?

The operator norm is important in matrix logarithm because it measures the size of a matrix and is used to determine if the matrix logarithm is well-defined and continuous. It also plays a role in the convergence of iterative methods used to calculate the matrix logarithm.

Can the continuity of matrix logarithm for operator norm < 1 be proven for all matrices?

No, the continuity of matrix logarithm for operator norm < 1 can only be proven for matrices with an operator norm < 1. For matrices with an operator norm > 1, the matrix logarithm may not be well-defined or continuous.

Similar threads

Replies
1
Views
1K
Replies
5
Views
1K
Replies
10
Views
2K
Replies
25
Views
3K
Replies
2
Views
3K
Replies
5
Views
2K
Replies
6
Views
2K
Back
Top