# Linear Algebra Proof - Determinants and Traces

• chill_factor
In summary, the problem is to prove that for an operator A, the determinant of e^A is equal to e^(Tr(A)). This can be solved by considering A in the basis of its eigenvectors, which is known as the Jordan form. This approach can be understood using basic linear algebra concepts, even for those with limited background in the subject.

## Homework Statement

Prove for an operator A that det(e^A) = e^(Tr(A))

## The Attempt at a Solution

I have no idea how to start. Can someone give me a hint?

In general the operator A represented by a square matrix, has a trace Tr(A) = Ʃ A (nn) where A (nn) is the nth row nth column. I don't know how to write the determinant in such a form that's understandable to myself and don't know how to compare them.

I have absolutely no background in linear algebra except a very elementary class on matrix arithmetic/eigenvalue problems and this is part of our "review" for a quantum mechanics class! Very few of my classmates have seen this math before and can't answer, I'm desperate!

Is this possible to solve this using very elementary steps? I've found an advanced proof that I have no idea how to understand. What is a Jordan form?

The basic idea is that you consider A in the basis of its eigenvectors. Is that enough of a clue?

voko said:
The basic idea is that you consider A in the basis of its eigenvectors. Is that enough of a clue?

Yes I figured it out. thank you greatly.