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An Intuitive Derivation of Eigenvectors

by dhruvp on 3/10/19, 2:47 AM with 26 comments

  • by daeken on 3/10/19, 5:39 AM

    If you want to get an intuitive, visual understanding of linear algebra -- including eigenvectors/eigenvalues -- 3blue1brown's playlist on the subject is just ... perfect. https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...

  • by platform on 3/10/19, 12:21 PM

    really good explanation. I like it even better than 3blue1brown or visualisations that I had seen.

    It is better because it really covers every step of the construction process.

    And offers explanation of why certain thing are not the right construction blocks. The author gives a visual example, for example, of why basic vectors 1,0 -1,0 are bad. The article shows they cannot span the whole space.

    Those kinds of explanations of 'bad constructions' are difficult to show in visualizations, that show 'good' constructions only.

    But, yet, in my view, these negative examples, are really helpful to explain the material that otherwise, requires 'intuition'.

    Not everybody has same intuition, so showing negative examples/impossible constructions, and why those do not work -- is a good way tuning one's intuition.

    ---

    On a separate note, I am wondering if such good step by step + counter examples, knowledge presentation -- is a result of author studying at MIT, or a natural trait (or both) ?

  • by FabHK on 3/10/19, 11:48 AM

    One way to think of eigenvectors:

    Your linear map A moves things around, and you aim to characterise the linear map.

    So, look for lines (through the origin) that are not moved. Those are given by eigenvectors. A point on that line might be moved closer to or further away from the origin (depends on eigenvalue < or > 1), or even flipped to the other side (if eigenvalue < 0), but the line as a whole is mapped to itself.

  • by quickthrower2 on 3/10/19, 11:37 AM

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  • by lxe on 3/10/19, 5:35 AM

    I also really liked this explanation: http://setosa.io/ev/eigenvectors-and-eigenvalues/

    You can drag things around and change values -- if you're a visual learner, it really helps grasp things like this.

  • by jules on 3/10/19, 3:53 PM

    IMO eigenvectors are easiest to understand in connection with differential equations, and that's also one of their most important applications. If you plot the flow of the equation x' = Ax then the eigenvectors are visually apparent.

    https://www.wolframalpha.com/input/?i=stream+plot+%7B-5x+%2B...

    The eigendirections are the directions where the solution moves in a straight line.

    Not all matrices have (real valued) eigenvectors:

    https://www.wolframalpha.com/input/?i=stream+plot+%7Bx+%2B+5...

  • by itissid on 3/10/19, 6:23 PM

    The difference between an exposition in text(like this one) and videos(3blue1brown) is how people prefer to build knowledge and more deeply with how one learns. The quality of both of these is excellent. And one can test what works best like explaining to oneself(or a rubber duck) after reading/viewing the material and how one can recall things.

  • by Koshkin on 3/10/19, 2:42 PM

    There is no shortage of intuitive explanations of various elementary concepts in mathematics (and physics); my personal favorites are by E. Khutoyansky [1]. (Surely enough, there is a video on eigenvectors!)

    [1] https://m.youtube.com/user/EugeneKhutoryansky

  • by rodionos on 3/10/19, 4:26 PM

  • by skywal_l on 3/10/19, 11:51 AM

    The thing I use to visualize an eigenvector is exactly that. A rotating planet. The eigenvector of the rotation matrix being the axis of rotation.

    It gets weird when thinking of 2D rotations though... Too complex for me!

  • by BeetleB on 3/10/19, 6:01 PM

    Good writeup! A tad bit disappointed as I was expecting some new insights, but this is more or less standard material about eigenvectors you would get in a typical university course.

  • by billfruit on 3/10/19, 4:54 PM

    I really think some better name than eigenvectors (and eigenvalues, etc..) should be popularized. I find them to be very obtuse and opaque terms.

  • by wittedhaddock on 3/10/19, 4:55 AM

    this is awesome tyvm!