|
> However, if v is a right eigenvector with eigenvalue \lambda, then v^T is a left eigenvector with eigenvalue \lambda, as well. Consider the matrix M = [ 0.50 0.50 ]
[ 0.25 0.75 ]
Clearly [1; 1] is a right eigenvector with eigenvalue 1. But [1 1] M = [0.75 1.25].> A matrix does not necessarily have nontrivial eigenvectors. Think about the 0 matrix here. The zero matrix has _all_ the nontrivial vectors as eigenvectors. > But, if A is nonzero, then it must have at least one nonzero eigenvalue, hence one nontrivial eigenvector. Consider N = [ 0 1 ]
[ 0 0 ]
This has no non-zero eigenvalues, but it is not the zero matrix. It does have a nontrivial eigenvector, [1; 0], because every square complex matrix has an eigenvector. (This is in contradiction to another statement you said, that a matrix does not have nontrivial eigenvectors—that is technically true but only in a limited sense that your space might not be ℂ^n or something that can be quickly generalized to it like ℝ^n. But like we’re in a computing forum and I might find that pedantic.)I don't know why you consider the eigenvalue zero to somehow not count as an eigenvalue. Very suspicious. > This also illustrates the conditions necessary for an nxn matrix A to have n linearly independent eigenvectors: the rows of A must be linearly independent. Consider P = [ 1 1 ]
[ 0 1 ]
The rows are linearly independent, but this does not have a second eigenvector.> All of this is covered in a decent undergrad linear algebra course. I would suggest either finding a video course, or getting a good book and working through it, if you want to understand these things better. sigh ... See this is why I sometimes feel bad about commenting here. Like, this comment thread was supposed to be a celebration of this different way of thinking about linear algebra, and then I have to deal with this stuff. Like I totally don’t think you meant to come across as condescending, but given that I have taught crash courses for struggling friends on linear algebra concepts they missed in their undergrad to get through our graduate work in physics, you know, it kind of does come across that way. |
For the first matrix, maybe a simpler example is M = [1 1; 0 0]. Then [1; 0] is a right eigenvector with eigenvalue 1, but [1 0] M = [1 1].