Concepts of Eigenvectors/values

[DrummingAtom]

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.

 

[Deveno]

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.

 

[HallsofIvy]

Once you have an eigenvalue, $\lambda$, for linear transformation, A, you find the eigenvectors by solving the equation $Ax= \lambda x$ 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.

 

[DrummingAtom]

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?

 

[CellCoree]

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.