Concepts of Eigenvectors/values

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Discussion Overview

The discussion revolves around the concepts of eigenvectors and eigenvalues, particularly in the context of a differential equations and linear algebra class. Participants express frustrations with the course structure and seek clarification on the procedures for finding eigenvectors after determining eigenvalues, including the implications of row operations on matrices.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant expresses dissatisfaction with their course, noting a lack of focus on linear transformations and diagonalization, which are relevant to understanding eigenvalues and eigenvectors.
  • Another participant questions the meaning of "switching the rows" and explains that changing the order of rows in a matrix does not affect the solution space of the corresponding linear equations.
  • A different participant clarifies that after finding an eigenvalue, eigenvectors can be found by solving the equation Ax = λx, and that row operations can be used freely in this process.
  • A participant notes that their eigenvector calculations often result in matrices with entire rows of zeros and wonders if this is related to the nature of eigenvectors being "stretched" along their lines.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the implications of row operations for eigenvector calculations, and there is uncertainty regarding the presence of rows of zeros in the resulting eigenvectors.

Contextual Notes

Participants do not fully explore the underlying assumptions about the matrices or the specific conditions that lead to the presence of zeros in eigenvectors, leaving these aspects unresolved.

DrummingAtom
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Just a small rant to start. My DE/LA class is absolute nonsense. I am really wishing that I taken each class separate because this class is making me and many others lose the big picture. For instance, this class doesn't teach linear transformations and diagonalization both of which I keep seeing anytime I look up stuff about eigens. I've managed to learn some of these on my own but I can't spend too much time on that stuff for now.

Anyway, after I find the eigenvalues am I allowed to do any switch rows to find the eigenvectors? Or do I have to account for the switching of a row in the eigenvector? Most of the matrices that are produced from plugging in the eigenvalues have some strange configurations, usually they have a ton of zeros and not what I'm used to. Thanks for any help.
 
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i'm not sure what you mean by "switching the rows".

what i WILL say, is that if a matrix A models a set of linear equations, changing the order of the rows amounts to changing the order that the equations are written in, and does not change the solution space.
 
Once you have an eigenvalue, [itex]\lambda[/itex], for linear transformation, A, you find the eigenvectors by solving the equation [itex]Ax= \lambda x[/itex] for x. IF you choose to use "row-reduction" to solve that equation, yes, you can use any row operations you wish to solve it.
 
Deveno said:
i'm not sure what you mean by "switching the rows".

what i WILL say, is that if a matrix A models a set of linear equations, changing the order of the rows amounts to changing the order that the equations are written in, and does not change the solution space.

Yeah that's I mean about switching the rows. That clears up that question.

Now, another question is why does every eigenvector I'm trying to find always seem to have an entire row of zeros? Is it because the eigenvector is capable of being "stretched" on it's line?
 

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