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.
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.
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?