Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

a) Prove that $(e^{At}-I)/t\rightarrow A$ as $t\rightarrow 0$, meaning $||(e^{At}-I)/t - A|| \rightarrow 0$ as $t\rightarrow 0$ for all $A\in\mathbb{C^{n \times n}}$. Hint: You may use the inequality $||A^k||\leq n^{k-1} ||A||^k$

b) For differentiable matrix functions $A, B: \mathbb{R} \rightarrow \mathbb{C^{n\times n}}$ we have the product rule $(AB)'=AB'+A'B$. What is the difference to the case of plain functions?

c) Assume the formula det($e^{A}$)$=e^{tr(A)}$ for all matrices $A \in \mathbb{C^{n\times n}}$. Show why this implies that the exponential always yields a regular matrix.

d) Challenge: Prove the formula mentioned in item c).

share|improve this question
2  
OK, Prof. Regina! –  Git Gud Feb 23 '13 at 21:19
    
Is this homework or part of some deep research investigation? –  copper.hat Feb 23 '13 at 21:21
    
I'm a Math's students thanks. –  Regina Feb 23 '13 at 21:21
1  
The point is that the idea of this site is to provide help, not do your homework for you. Show some work, indicate what you are having problems with, etc... –  copper.hat Feb 23 '13 at 21:22
1  
Hints for (a),(b),(c),(d): It might help if you actually use the definition of the matrix exponential. –  wj32 Feb 23 '13 at 21:53

1 Answer 1

(d) is the only cool question of the bunch. Here is my take on it:

1) Formula (c) is easy to prove for triangular matrices.

2) Since $$\det(e^A)$$ and $$e^{tr(A)}$$ are invariant under basis tranformation it follows from (1) that it holds for any matrix. (Think about Jordan Normal Form).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.