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Insights What Are Eigenvectors and Eigenvalues? - Comments

  1. Jan 22, 2016 #1

    Mark44

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  2. jcsd
  3. Jan 22, 2016 #2

    QuantumQuest

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    Great job Mark!
     
  4. Jan 22, 2016 #3

    RJLiberator

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    Excellent information, Mark44!
     
  5. Jan 22, 2016 #4

    Mark44

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    Thanks! My Insights article isn't anything groundbreaking -- just about every linear algebra text will cover this topic. My intent was to write something for this site that was short and sweet, on why we care about eigenvectors and eigenvalues, and how we find them in a couple of examples.
     
  6. Jan 22, 2016 #5
    I learned about eigenvalues and eigenvectors in quantum mechanics first, so I can't help but think of wavefunctions and energy levels of some Hamiltonian.

    Other cases are (to me) just different analogs to wave functions and energy levels.

    Nice article. I eventually took linear algebra and learned it the way you are presenting it, but I was a senior in college taking the course by correspondence.
     
  7. Jan 23, 2016 #6
    So the red/blue arrows on the image are eigenvectors?
     
  8. Jan 23, 2016 #7
    Last edited by a moderator: Jan 24, 2016
  9. Jan 23, 2016 #8

    Mark44

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    What image are you talking about? The article doesn't have any images in it.
     
  10. Jan 23, 2016 #9

    Samy_A

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    It's a mystery challenging the basic foundations of Physics: he seems to refer to the image mentioned in the post following his own post. :oldsmile:
    If that is the case, the blue and violet arrows are the eigenvectors, not the red.
     
  11. Jan 24, 2016 #10

    Krylov

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    Could it be a reference to Mona Lisa at the top of the Insight?
     
  12. Jan 24, 2016 #11

    WWGD

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    How about asking Harouki to write an insight on that one -- time travel??
     
  13. Jan 25, 2016 #12
    i do not understand how det(A-lambda(I))=0
    since x is not a square matrix we cannot write det((A-lambda(I))*x)=det(A-lambda(I))*det(x)
     
  14. Jan 25, 2016 #13

    Samy_A

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    Correct, you can't write that. Note that Mark44 doesn't write ##|A – \lambda I||\vec x| = 0##. He correctly writes ##|A – \lambda I|=0##.

    In general, if for a square matrix ##B## there exists a non 0 vector ##\vec x## satisfying ##B\vec x=\vec 0 ##, then the determinant of ##B## must be 0.
    That's how ## (A – \lambda I)\vec{x} = \vec{0}## implies ##|A – \lambda I|=0##.
     
  15. Jan 25, 2016 #14
    but howw???
    ah i got it just as i was writing this.
    let 'x' be non a zero vector and let det(A) ≠ 0 .
    then premultiplying with A(inverse) we get
    (A(inverse)*A)*x=A(inverse)*0
    which then leads to the contradiction x=0
    am i right???
    i'm sorry that i don't know how to use latex.
     
  16. Jan 25, 2016 #15

    Samy_A

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    Yes, that's basically it.
     
  17. Jan 28, 2016 #16
    Nice insight !!!
    If you like it, I have an exemple of application for your post to euclidean geometry. You could explain how eigenvalues and eigenvectors are helpfull in order to carry out a full description of isometries in dimension 3, and conclude that they are rotations, reflections, and the composition of a rotation and a reflection about the orthogonal plane to the the axis of rotation.
     
  18. Jan 28, 2016 #17

    Mark44

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    Thank your for the offer, but I think that I will decline. All I wanted to say in the article was a bit about what they (eigenvectors and eigenvalues) are, and a brief bit on how to find them. Adding what you suggested would go well beyond the main thrust of the article.
     
  19. Mar 5, 2016 #18
    For how this helps the physics people, the eigen values reduce the components of a large tensor into only as many components as the order of the matrix. These reduced ones have the same effect as all the tensoral components combined. About eigenvectors, I'm not sure how it is applied.
     
  20. Apr 1, 2016 #19

    ibkev

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    Eigenvalues/vectors is something I've often wanted to learn more about, so I really appreciate the effort that went into writing this article Mark. The problem is that I feel like I've been shown a beautiful piece of abstract art with lots of carefully thought out splatters but the engineer in me cries out... "But what is it for?" :)

    "Here is an awesome tool that is very useful to a long list of disciplines. It's called a screwdriver. To make use of it you grasp it with your hand and turn it. The end." Nooooo! Don't stop there - I don't have any insight yet into why this tool is so useful, nor intuition into the types of problems I might encounter where I would be glad I had brought my trusty screwdriver with me.

    I would truly love to know these things, so I hope you will consider adding some additional exposition that offers insight into why eigenstuff is so handy.
     
  21. Apr 3, 2016 #20

    Mark44

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    My background isn't in engineering, so I'm not aware of how eigenvalues and eigenvectors are applicable to engineering disciplines, if at all. An important application of these ideas is in diagonalizing square matrices to solve a system of differential equations. A few other applications, as listed in this Wikipedia article (https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) are
    • Schrödinger equation in quantum mechanics
    • Molecular orbitals
    • Vibration analysis
    • Geology and glaciology (to study the orientation of components of glacial till)
     
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