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Textbook to help me understand eigenvectors and diagonalization

  1. Jul 23, 2012 #1
    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
     
  2. jcsd
  3. Jul 23, 2012 #2
    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.
     
  4. Jul 23, 2012 #3
    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.
     
  5. Jul 23, 2012 #4
    Well, what exactly is causing your confusion? How a matrix can be diagonalized, what an eigenvector is, how we find them?
     
  6. Jul 23, 2012 #5
    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?
     
  7. Jul 23, 2012 #6
    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.
     
  8. Jul 24, 2012 #7
    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!
     
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