Eigenvectors and eigenvalues | Essence of linear algebra, chapter 14

Eigenvectors and eigenvalues | Essence of linear algebra, chapter 14

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  1. I actually think courses over-prepare for this topic; so by the time the concept is introduced (always with an example, never a succinct definition), the student sees the familiar identity matrices, diagonals, vector multiplications, etc. and says, "So what is it that makes eigen-whatevers so special?" Fortunately, an active graphics presentation is perfectly suited to the task of clarification.

  2. It's so funny to see those German terms in your English! Hilarious! One explanation: "Eigenvector" means self-vector and "Eigenvalue" mean self-value….. I hope some people get a clue about those terms now…. 😉

  3. This video gave me so many "AHA!" moments and cements all the information you've taught in former videos of the series. Thank you so much!

  4. the sandwiched part though, why is it true? why does sandwiching a matrix by both its (eigenvectors put into a matrix?) and the inverse of the (eigenvectors put into a matrix?) result in doing the same transformation as the initial matrix but in a different coordinate system with eigen vectors as the basis of said coordinate system?

  5. Excellent I have no idea what change of basis etc means except that as I work with animation graphics all your animations seem familiar. Your diagrammatic explanation of Eigenvectors and eigenvalues made perfect sense as these concepts are also familiar as the scaling along a none global axis. That is a very good video and excellent communication of a concept even for a none mathematician. Making animations I should not use negative scales but sometime it is useful 🙂

  6. I wish I had seen these videos before I took quantum mechanics in college. I would have had a much, much easier time.

  7. Thanks for the great video!

    For complex eigenvalues of a real matrix, its eigenvectors are also complex (and its conjugate should also be the other eigenvectors). I am wondering if there is a geometric way of understanding such eigenvectors like we did for real number–scaling case.

    I noticed this video also from 3B1B saying that 'i' indicates rotation from real axis to imaginary axis:

    Does these two rotations coincide or actually there is a inner link between these two?

  8. Do you think you could make videos on SVD and Moore-Penrose Pseudoinverse as additions to the series? It would be great to have visual representations of those two topics, I find SVD way harder to visualize than eigendecompositions. Thanks!

  9. The transformation is real and meaning-laden. But the solution to finding an eigenvalue is rather mathematical trickery – in that this Squishification does not really happen, it is just a mathematical means of solving for zero. So, mid-stream in the explanation we switched from an intuitive approach (conforming to my logical reconstruction of what is being said) to a "logically unsound assumption" of turning the two-dimensional into a one-dimensional in order to effect a solution (after which, we just ignore the "fact" that we are left in a one-dimensional world and suddenly for no apparent reason return into and start talking about a two-dimensional world). This use of an abstraction should be clearly delineated as a "bridge of abstraction", a mathematical trickery tool to more easily effect an answer without in anyway implying that this state of affairs has reality or any meaning on the transformation. If I have this correctly.

  10. I was a math major and pulled A's in linear algebra, and yet only after seeing this video did I look at the formula for the eigendecomposition of a matrix and say "well, that's obvious." Goes to show that there's always more to grasp by getting a different view of the topic.

    (And if you're reading this and don't know what the heck I just said, all the eigendecomposition means is that if a matrix has enough eigenvectors to form a basis, you can represent that matrix as just a scaling transformation _in that basis_.)

  11. Grant, please extend your linear algebra series to include the concepts of extensively applied operations such as SVD and LU decompositions and how they manifest in the vector space. There are many out here like me who would love to visualize and appreciate what we do in practice.

  12. 11:34
    The mathematician that first came up with "imaginary" numbers refered to them as lateral numbers, as saying a number that doesnt belong to the Real numbers plane. I'm pretty sure that in this case, the eigen vector is the one that pops out of the screen, the axis of rotation, just as in the 3d model.

  13. This is great. When I was in school doing this I found the computation ok and I understood conceptually what I was doing but we were really provided no visual understanding of what was happening. All I remember now about eigenvaues is that I had to write a lot of lambdas.

  14. @3blue1brown Please tell me what book should i read to get all understanding of linear algebra visually, as you show. I'm a beginner cum intermediate, still tell me which book should i follow??

  15. Mathematical concepts evolved as the contemporary mathematicians worked at problems of their times but in school people are told things mechanically without attempting to xplaining the evolution of concepts thru solving problems
    That is y we generally dont understand y v do things in school xcept for clearing exams

  16. I'm here for the glorious next level math animations. I love your channel man. Thanks for expanding my mind!👌
    INSTANT upvote.

  17. I always appreciate your brilliant videos about linear algebra. However, I have no any idea in visualizing Cayley hamilton thorem. Would you explain them for next video?

  18. Where did you learn all this? I mean, I never had a teatcher nor found a book that could explain linear algebra this way. Where do you find all these concepts? I wish I had had access to all of this years ago it would have been a lot more interesting than abstract definitions that seemed meaning less at the time. Thank you

  19. Wow !! LOVED IT. Sir, you give maths a meaning, similar to the quote you mentioned. It is way way way more than calling it just a manipulation of digits. I feel lucky to be able to explore such plethora of knowledge. I am really grateful that you shared this with me.

  20. Not sure if this is a correct way to interpret this, but the result of the eigenvector from a rotation made me think of an imaginary plane going through the screen (z-plane). Seen this way, the eigenvector i makes a lot of sense since it is the axis of rotation!

  21. Can any math wizard here please help me understand the concept that – "Why Eigenvector of Covariance matrix points into direction of maximum variance of the data." I am finding it difficult to understand it intuitively.

  22. Oh God that exercise at the end! It was SO GOOD hahahaha I had lot of fun tonight, I'm surprised because the eigenvalues in there generated a lot of non-integers, and it seems unbelieveable that it makes sense and ends up to be all integers in the powers of the matrix, and such that that pattern applies. Also, being able to demonstrate the pattern was soooooooo satisfying!

  23. The statement "there could be no eigenvectors" is clearly wrong. It will be ok to say that some parts are too advanced for this course. But saying its not exists? It makes people belive that all of this is simple, but it is actually not so. It is the reason why some university lectures are not so popular. They are just honest.

  24. Help needed ! I understand that only a subset of 2D-transformations have 2 eigenvectors. Could someone explain how to visually characterize these transformations please ? Is it correct to say they are only those consisting in streching in one direction, and then stretching in another direction ? I'm not sure. What happens if I stretch a 3rd time in yet another direction ?

  25. Hello…
    I really appreciate what you're doing in this course… extremely useful for curious people
    Please make a set of videos on Metric space and non linear transformation..
    Please do respond for this …
    With regards..

  26. If you have a classic course on linear algebra, then hate it as almost everyone does, then spend a few years without even thinking about it, then realizing your life would really improve if you had learned it and then watch 3blue1brown videos and after that go back and study linear algebra again (by books or online courses), then I think you will master it lol

  27. In all my years of uni I’ve never understand the point of eigenvectors. I’ve known how to calculate them just never knew why I had to!

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