Concepts of Eigenvectors/values

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In summary, The speaker is frustrated with their combined DE/LA class and wishes they had taken them separately. They have learned about eigens on their own, but are unsure about using row-switching to find eigenvectors. The other person explains that switching rows does not affect the solution space and that eigenvectors can have an entire row of zeros due to stretching.
  • #1
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.
 
  • #3
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.
 
  • #4
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?
 
  • #5


I understand your frustration with your current DE/LA class. It is important to have a clear understanding of the fundamental concepts in order to fully grasp the application of eigenvectors and eigenvalues. Linear transformations and diagonalization are indeed key concepts in understanding eigens, and it is unfortunate that your class does not cover them.

In regards to your question about switching rows to find eigenvectors, it is important to note that eigenvectors are not affected by row switching. This is because eigenvectors are defined as non-zero vectors that are only scaled by the corresponding eigenvalue when multiplied by a matrix. Therefore, you do not need to account for row switching when finding eigenvectors.

It is also common for matrices produced from plugging in eigenvalues to have unusual configurations, such as a large number of zeros. This is because eigenvectors are essentially "special" directions in which the matrix only stretches or shrinks, without changing direction. This can result in matrices with many zeros, as other directions are not affected by the eigenvector.

In conclusion, while it may be frustrating to not have a comprehensive understanding of eigenvectors and eigenvalues in your class, I encourage you to continue learning and exploring these concepts on your own. They are fundamental to many areas of science and mathematics, and having a strong understanding of them will greatly benefit your future studies and research.
 

1. What are Eigenvectors and Eigenvalues?

Eigenvectors and Eigenvalues are mathematical concepts used to describe certain properties of a linear transformation or a matrix. Eigenvectors are vectors that do not change direction when a linear transformation or matrix operation is applied to them, but only change in magnitude by a scalar factor. Eigenvalues are the corresponding scalar values that represent the amount by which the Eigenvector is scaled.

2. How are Eigenvectors and Eigenvalues used in data analysis?

Eigenvectors and Eigenvalues are commonly used in data analysis to reduce the dimensionality of a dataset. By identifying the Eigenvectors and Eigenvalues of a dataset, it is possible to reduce the number of variables needed to represent the data, while still retaining most of the essential information. This process is known as Principal Component Analysis (PCA).

3. What is the significance of Eigenvectors and Eigenvalues in quantum mechanics?

In quantum mechanics, Eigenvectors and Eigenvalues are used to describe the state of a quantum system. The Eigenvectors represent the possible states of the system, while the Eigenvalues represent the corresponding energy levels. The concept of superposition, where a quantum system can exist in multiple states simultaneously, is based on the idea of Eigenvectors and Eigenvalues.

4. Can Eigenvectors and Eigenvalues have complex values?

Yes, Eigenvectors and Eigenvalues can have complex values. In fact, in quantum mechanics, it is common for Eigenvectors and Eigenvalues to have complex values. This is because quantum systems can exist in a superposition of states, and complex numbers are needed to represent this phenomenon.

5. How are Eigenvectors and Eigenvalues calculated?

The process of calculating Eigenvectors and Eigenvalues involves solving a system of linear equations. The Eigenvectors are obtained by finding the null space of the matrix, while the Eigenvalues are the solutions to the characteristic equation of the matrix. In some cases, numerical methods may be used to approximate the Eigenvectors and Eigenvalues for large matrices or complex systems.

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