Commutative linear operators and their properties

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Homework Help Overview

The discussion revolves around the properties of commutative linear operators, particularly focusing on their eigenvectors and the conditions under which they can be simultaneously diagonalized. The original poster seeks clarification on these concepts and their implications in linear algebra.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to explore the relationship between commuting linear operators and their eigenvectors, questioning the possibility of simultaneous diagonalization and the proof of such properties. Other participants inquire about specific examples, such as 2x2 matrices that may not be diagonalizable.

Discussion Status

The discussion appears to be in its early stages, with some participants expressing the need for assistance and others attempting to redirect the conversation to a more appropriate forum. There is an indication of interest in exploring the topic further, but no definitive answers or consensus have emerged yet.

Contextual Notes

There is a mention of a potential misunderstanding regarding the appropriate forum for this topic, which may have affected the level of engagement and responses received.

McLaren Rulez
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Can someone help me with this? When two linear operators commute, I know how to show that they must have at least one common eigenvector. Beyond this fact, what else can be said about commutative operators and their eigenvectors? Further, can they be diagonalized simultaneously (or actually, can they be diagonalized in the first place?), and if so, how can this be proved?
 
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I wished to know about this but no one is responding to this thread. Please, can one somebody help?
 
kthouz said:
I wished to know about this but no one is responding to this thread. Please, can one somebody help?

Well, what have you thought about so far? Take 2x2 matrices. Can you find one that can't be diagonalized?
 
kthouz, I asked this question again at https://www.physicsforums.com/showthread.php?t=501340

It was a mistake on my part as I had posted this in homework section instead of linear algebra where it belongs. That's why you are not getting replies. If you have a different question from what I asked in the link, try posting a new thread in the linear algebra section. You will definitely get help
 

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