Linear Algebra Proof - Determinants and Traces

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Homework Help Overview

The problem involves proving the relationship between the determinant of the exponential of an operator A and the trace of A, specifically that det(e^A) = e^(Tr(A)). This falls within the subject area of linear algebra.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the use of Jordan form and the basis of eigenvectors as potential approaches. Some express uncertainty about how to start the proof and seek hints or clarification on concepts like the trace and determinant.

Discussion Status

The discussion includes various attempts to clarify the concepts involved, with some participants offering guidance on considering the operator in the context of its eigenvectors. There is a recognition of differing levels of background knowledge among participants.

Contextual Notes

Some participants indicate a lack of familiarity with linear algebra concepts, noting that this problem is part of a review for a quantum mechanics class. There is mention of advanced proofs that are difficult for some to understand.

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Homework Statement



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

Homework Equations





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.
 
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How about A's Jordan form?
 
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.
 

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