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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).

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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
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
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).

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