Textbook to help me understand eigenvectors and diagonalization

[tamtam402]

Hi, I'm currently self-teaching myself some mathematics needed to study physics. I'm working through the book Mathematical Methods in the Physical Sciences by Mary L Boas. The book is a well known one, and it's used in many physics programs to teach their math courses.

However, I've read the section on eigenvectors and diagonalization many times, and I still feel like I don't really understand what's going on.

Is there a book (or even better, a free online textbook) where these subjects are explained in a clear way? I don't want a very abstract book destined to math majors, I'd like something written for physics/engineering students.

Thanks in advance

 

[Jorriss]

Do you know what is causing your confusion?

First off, you could always just look at a pure linear algebra text like Strang, Anton or Lay.

Second, you may want to look at a quantum mechanics book and see those topics in a physical context. A classical mechanics book would also have it when discussing rotational dynamics of rigid bodies.

 

[tamtam402]

The confusion mostly comes from the fact that very few explanation is given in the book, which is a consequence of the vast amount of material covered. Somehow I don't feel like eigenvectors/diagonalization can reastically be exlpained in 5-6 pages.

I'll try to find one of the books you listed at my university's library, thank you.

 

[dydtaylor]

Well, what exactly is causing your confusion? How a matrix can be diagonalized, what an eigenvector is, how we find them?

 

[tamtam402]

I understand how to diagonalize, how to find eigenvalues and eigenvectors, but I don't understand what they're used for. I know that the vectors in the first system that are parallel to the eigenvectors will simply be shrunk/extended without being rotated or reflected, but that's pretty much it. What's the point of finding the eigenvectors?

 

[dydtaylor]

From what I can tell (speaking from probably about a year more experience than what you have, so there are most likely some details missing), one of the main benefits of diagonalization is it allows you to simplify an operator; instead of having a very messy matrix, you can simplify it greatly and find it's eigenvalues very easily.

Eigenvalues and vectors have a lot of applications. For differential equations, if you have a system of differential equations then you can represent the system through a matrix, and find solutions from the eigenvalues/vectors of the matrix (simplified greatly, but more detail here http://tutorial.math.lamar.edu/Classes/DE/SolutionsToSystems.aspx)

As for other applications, eigenvalues and eigenfunctions of operators give measurable values in Quantum Mechanics (note that functions are vectors in Quantum, but to understand why takes a little more theoretical understanding of Linear Algebra).

For a more mathematical application, if you have a diagonalizable matrix, you can use eigenvectors to create a basis for the vector space you're working with (and vice versa; they're equivalent) this is nice when working in a new mathematical system. For instance, in quantum we use this to help find all of the possible solutions to the 1-dimensional harmonic oscillator.

 

[tamtam402]

Thanks for the explanations, and for the link. On that website I found paul's notes on eigenvectors and they were much clearer than my book!