How Do Eigen Vectors and Eigen Values Change with Dimensionality?

[esash28]

Hi all,
I am a complex systems researcher and I need to have complete knowledge about eigen vectors and eigen values. How does change in dimension affect a point's eigen vector and eigen value? What does principal eigen vector and principal eigen value mean for a point of n-dimension?

Please help.
Thanks in advance.
Esash

 

[Fredrik]

If you need complete knowledge about them, you're going to have to read a book on linear algebra. I like "Linear algebra done right" by Sheldon Axler. (If it's not your style, you can find other recommendation in the science book forum, in the academic guidance section). Very briefly, if V is a vector space, and T:V→V is linear (i.e. satisfies T(ax+by)=aTx+bTy), a vector v is said to be an eigenvector of T if there's a number a such that Tv=λv. The number λ is called the eigenvalue associated with the eigenvector v. For example, if V is $\mathbb R^3$, and T is a rotation in the xy-plane, any multiple of (0,0,1) (i.e. any vector in the direction of the z axis) is an eigenvector of T with eigenvalue 1.

I don't know what a principal eigenvector is. I tried looking it up at Wikipedia but I get redirected to "eigenvector", where the term is used, but not explained. The text under one of the images said something about T transforming all vectors toward the principal eigenvector, but it can't be true in general that there's an eigenvector with that property.

 

[esash28]

Principal eigen vector is the eigen vector corresponding to the largest eigen value.

I use the book, Introduction to Linear Algebra by Gilbert Strang. I know mathematically what it all means. But, I become blank when it all comes to explaining things, non-mathematically or physically. So please help.

Thanks in advance.
Esash

 

[Fredrik]

I think you will have to ask more specific questions. Right now it sounds like you want someone to rewrite a book in a style that suits you better.

 

[esash28]

Well, I need some guidance about how to learn what eigen vectors and eigen values are, in a non-mathematical way. What are they? Why are they so useful? How do they determine the critical properties of nature? In fact, Everything about them. Is there any such descriptive tutorial?

 

[elibj123]

I think that as a researcher you will have to dive into mathematics to learn this.
It'll begin with some linear algebra and the subject of operators (transformations), and will continue with the study of dynamical systems and their analysis, that is, differential/difference equations.

While one can certainly write out a list of rules and conditions to follow schematically, or even by a computer program, as a "researcher" you'll need much more than this. That's what separates a good academic who lingers on understanding the subtle notions and there implication on his field of work, from just a guy who plays around with numbers, entitling himself of something he doesn't really deserves.

That is why, you will have to study some maths to really understand something (and it won't be everything). In the course of learning you will find many answers to your questions (what are they? why are they so useful?) and beyond.

 

[republicofmat]

The Wiki article seems pretty good to me:

http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace