You touched on linear algebra, differential equations, and probability/statistics, three of the four areas of math that underlie most engineering and computing applications today (the fourth being number theory.) Nice video.
You and grant Sanderson are the reasons my interest in "understanding" mathematics is still alive ... I just can't express in words how much I owe to you both 🙏🙏🙏huge respect from India
This was a better explanation in 20 minutes than anything in my Diff Eq + Lin Alg class. I just finished a test on eigenmagic with a decent grade, and I didn't know what an eigenvalue even was.
Great quote from the video, “Eigenvectors/Eigenvalues (in many cases) reveal long term behavior.” Why is this not the leading sentence for every math class discussing this topic? All of the sudden its application became extremely clear.
Having watched all the previous videos, this one felt a lot like a sequel and I simply loved it. As a Math teacher having to teach Linear Algebra to engineering students, your videos are a great source of inspiration! Thank you!
Dawn, you can turn yourself into an engineer just by watching this and other few channels on youtube (believe me, I'm an engineer). It's so well explained and easy to understand that makes real classes tiring and boring. Thank you so much for spending your time making those helpful videos!
@@BooyahLThat's true, but most of the concepts you can find here on youtube, even some applications/projects and advanced exercises. I know it doesn't replace a real engineering course, but there are so many things that you find here on youtube that if you had a guide of what to study next, you would have a pretty good idea of what is a engineering course.
I used to think that too but lately I’ve changed my stance. Learning is iterative and I think what makes videos like these so good is normally because this isn’t your first pass over the subject
No joke, your content has singlehandedly revived my interest and passion for pursuing both learning and teaching math and algorithms. I’m going to really, finally, put my nose to the grindstone to cop an ICPC medal off this inspiration high. You’re the best, Zach, and you have a new Paterson member. Godspeed!
So, I'm from Uruguay (Our native language is Spanish), and I was hearing the video (Started with a related one, of the same channel) and thought: "Ok, eigenvalues, and eigenvectors are really cool! So why I'm not learning this at University?", And I searched the translation, that is: "Valores propios, y Vectores propios" respectively. I'm shock. I have been calculating the eigenvalues, and the dimensions of the subspaces associated, all over the place, but no one has ever teached us what the hack they are used for! Yeah... I'm probably dumb because I didn't searched about this before, and, I know, some times is hard to teach "all the aplications" of an unknown tool, (for the studient). But, is just far more engaging and useful to understand why i'm using "det(A-xI)" and solving a weird polinomium. Anyways. Thank you SO much, for the video.
This is very true. In physics, and especially in Quantum Mechanics we use eigenvectors/values all the time. It's really fundamental - but my course (many, many years ago) never once covered what they are geometrically or even in some depth mathematically. They were simply presented as a series of algebraic manipulations of arrays of numbers to get 'the answer'. To this day, even though I can calculate the probability amplitude of spin states of electrons (for example) I haven't the faintest idea why this model works as it does and how it came to be invented in the first place.
Hope you guys enjoy! Two things I want to mention 1) This video (mostly towards the end) references some things I discuss in the applications of matrices video ( ua-cam.com/video/rowWM-MijXU/v-deo.html ). If you haven't seen it you'll be fine for this video, but a few things will make more sense if you have seen it. 2) Here's the video where I link and explain the desmos file I used for this one. ua-cam.com/video/1_fXgJWQhVY/v-deo.html
School isn't killing interest in sciences. It just fails to keep up/compete with the insane amount of interactive, appealing and sometimes addictive content available online.
Daily life is THE perpetual school of sciences, be it material science (build a house, make a dress), math science (budget, electricity), language science (talk to your neighbor of dissimilar roots) or history science (voting), and only peer pressure is killing interest in being interested, therefore you, alone, are responsible for keeping your interests alive. I'm sure you will go far, maybe one day you will be in a position to direct the interests of other, to catch someone who is falling to peer pressure and show them the benefit of joining you in the journey.
@@b_kind force-feeding subjects instead of giving insights about applications is a frequently complained part of schools though, even on subjects that are very useful. That probably contributed to the "killing"
I had the pleasure of being directed to your videos today. Thank you tech companies (who paid attention in Linear Algebra classes) for using what I suspect to be a low-rank matrix decomposition to learn about my viewing preferences.
When the eigenvalue is imaginary, that means you got out of R² , and the eigenvector is perpendicular to the plan, that is the 3D vector [ 0 0 1 ], it's actually the rotation axes for the 2D rotation, that's our eigenvector in that case..., this also happens when solving vectors in 1D and getting the value “i” as result, that's actually the vector [ 0 1] in 2D => the vector perpendicular to the subspace we are in
I was thinking in similar lines since the imaginary plane is just the other plane of physical value that affects the current observed value we are dealing with. it's just in a different plane. Nice one.
Thanks you very much, this completely boosted my understanding after studying this a year ago and not knowing any practical applications or even the essence of studying and understanding them...i'm grateful for time and effort you took to make such a clear video...
Been binge-watching your transformation video. Although i've done my signal processing course (which i suck at), your video gave me new insights on how to look at those transformations, big thanks! :D
Dec 12: Me: finishes finals for differential equations and linear algebra. Dec 13. Major Prep: Eigenstuff Me: well, the timing couldn't have been better
When I got to the part about stability and rotation I was like: “Hey Laplace Transform!”. I love the way so much of math overlaps!
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Feedback: The last videos have been awesome! They speak to the intuition, and allow people a much better understanding of how things work and when they apply. You've done a fantastic job - thank you for doing it. This is the kind of video people should keep returning to until the info it contains stick - it is integral to understanding pretty much any complex system. This really helps! 👍
This was really helpful. I'm watching several videos on eigenvectors and eigenvalues right now because I did have this stuff in college, but back then, it wasn't well explained as it is with the animations you roll back and forth. I really appreciate this. Thank you.
We spent like 2 weeks learning about these in my math paper, I had no idea what any of it was about, but at least I could do the arithmetic. This video's visual explanation is soo much better for understanding what these are than any math text book
watching this now after just finishing my bachelors in mechanical engineering and going onwards to a masters in computational engineering. feels like the most amazing application-based review video. THUMBS UP!
4:50 note: to find the eigenvalue (1+sqrt5)/2, you use the equation (1-λ)(0-λ)-(1)(1)=0. to get this you set the right side equal to λ(F_(n-1) F_(n-2)) where λ is the eigenvalue, then get everything to one side. now you have a homogeneous system of equations. the matrix of coefficients for any homogeneous system has a zero determinant due to the right column of zeroes. use this to set up a quadratic equation and solve for λ using the quadratic equation
I understood the geometric interpretation of eigenvalues/vectors pretty well, and had a working knowledge of the applications in DEs and things like PCA, but without a real intuition for why it worked - this is the first video I've seen that really connected them for me, thanks.
I've been an engineering student in various forms for 5 years and I've studied the mechanics of calculating Eigenvectors and associated values for hour and hours. I've searched and searched for a 'simple' explanation of what an Eigenvector actually is. This is by far the best explanation I've found.
I had a brilliant doctoral computer science professor this semester and he pretty much told us that physicists and engineers he collaborated with over his career most of the time wanted fast ways to solve for eigenvalues on large computer systems lol
I just took my linear algebra final, and this video answers the question my professor failed to all semester: what is the point of all these techniques I'm learning?
@@SpaghettiToaster I'm a computer science major, and this is a requirement. If I wanted to study something for it's inherent beauty, I'm be a liberal arts major.
@The Thunderblade Damn dude, I was joking, but where at all did I say say anything approaching "what is maths?" Regardless of the subject, a student's going to grasp the material better with some understanding of context. For example, Mondrian's famous abstract art seems pretty simple if you see it out of context (you know, the ones with the thick black lines and geometric shapes in primary colors), but when you appreciate what the guy was able to do earlier in his career, and why he chose to go abstract instead, it lends a better understanding of why his art is so impactful. Check it out, I have no problem with learning advanced maths. My issue arises when I, along with a good chunk of my class, were asking in good faith, "what do these numbers we're coming up with mean?" I get that there are people who enjoy doing math just to see the formulae churn out answers, as that's who my LA professor was, but it's not me. This guy was teaching at a superficial level of just showing us various techniques without bothering to give us any insight into the larger implications of what the math meant, and it made the course more of a headache than it needed to be. Maybe go talk to someone about unleashing that kind of hostility on a total stranger, though. Can't be good for you to carry around that level of vitriol.
@@SpaghettiToaster I thought the question quite valid and mirrored my own when reflecting upon how and why the curriculum I followed was created. There must have been a perceived context within which what was to be taught was decided. As an example, although I did Linear Algebra we didn't touch upon Eigen 'stuff', however in a Numerical Analysis we were shown how to calculate the eigenvalues of n-dimensional matrices. I did wonder why. It was only after I graduated and taught myself Quantum Mechanics, starting with The Copenhagen interpretation that I came across Eigen 'stuff' and the penny dropped as to why. Mathematics in and of itself is beautiful but in a course created by a random subset of planetary mathematicians it can be useful to know the why because that hints towards the collective understanding of the Department's research staff and 'the arrow' of their vision. One always be brave enough to ask 'why' and not just settle for being told.
Thanks man. You have my respect. I have taken courses on linear algebra in my college, but the way you explain won my heart. This motivates me to learn about it more, whereas the studies in college were mostly numerical and boring. Thanks for all knowledge you are contributing to this world! Keep making such videos. I love your channel.
Wow! I'm a machine learning engineer, and I have had studied all these concepts time and time before in the college, but I had never grasped the gist of it this good, really surprising! thanks!
Great video ... one of the gold nuggets of UA-cam ... first time I saw some real applications (just was number crunching to me before this). Thanks ... it is most appreciated!
This is an excellent demonstration. I've been struggling with the concept of the Eigenvector for some time and have explored many sources for different explanations. This is by far the best, I think I may almost understand their purpose now. Please keep up the good work!
Fun fact for the mathematically inclined: This demonstrates a nice transition of 'Pure Math' theory followed by its 'Applied (Math)' to real world problems &/or situations....it's ALL about PRACTICE & MORE PRACTICE friends! Superb tutorial btw!
As a new student who can barely factor polynomials, this is a fascinating look into a topic I don't understand and can't perform yet. Thanks for making it digestible for those of us in our early struggles. These videos make me want to push harder and get myself to the point where I can understand and apply the concepts.
i appreciate this video. seeing real-world applications by using abstract math is one of the most profound things i have witnessed. knowledge is the real power behind being effective in the world. thank you for educating us. it makes the world a better place. (and a less ignorant place)
Your video is pure joy for me as a math lover. Im an engineering student( graduate level) with specialisation in control systems theory. I've seen many explanation videos, but yours is my favourite. If your video had been online while i took my math courses in undergraduate level, i could have saved a lot of hours :D. You have done a great job!
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These are the videos that make UA-cam great! Please continue to make this type of videos, I absolutely love them
As a postgraduate student in engineering, I sincerely recommend this video to all engineering students as most of our modules make use of matrix and eigenvalues! And this video really answers the question that I always have in mind "why characteristic polynomial and eigenvalue are so important"
Man this is brilliant. I saw this video about 4 months ago. Lately been going through matrix n clustering in python and it hit me and I rushed back to this vid. Everything makes perfect sense now. Am happy it clicked 🚀
This video is mindbogglingly good. Outstanding!!! You will have a new Patreon supporter shortly. I might have taken a different career path if I had watched these videos earlier in life. Outstanding.
love these videos you've made on linear alg. Im a soph ME student finishing up a course on lin alg and I've found your videos really entertaining, clarifying and well produced. Ever considered doing a video on basis vectors? properties and apps of the determinant? This is such useful math and I just wish my teachers could make these abstract topics as palatable and engaging as you have.
Your videos are really helpfull as they graphicly and schematicly describe mathematic and physic concept that other way would be almost imposible to understand, thanks Mr. Zach for the time and effort you put on making and posting your videos on UA-cam for the public. Me and lots of other people really appreciate it.
Someone may have already mentioned this, but the idea of transitions and equilibrium ties directly to the concept of limiting probabilities in a system - particularly in Markov Chains, under the right conditions. I find the connection to eigenvalues and eigenvectors in this context fascinating, and I really regret that we didn’t explore this aspect in my university course on probability and stochastic processes!
Zach, this is a great video, just in time to my exams, now i unsderstand the perpose and how much usefull are eigenvectors and eigenvalues, colleges dont teach math in the way they should, they dont explain "why is that important?" they just give u a book. U can find motivation watching this kind of videos, congrats!
Thank you so much, your explanation is simple and clear and contains alot of concepts which makes the video not boring(which math videos usually are) and intriguing
Good video, thank-you. As a child I was part of a pilot experiment that taught 'modern maths', (SMP 1-5), from 11-17 yrs old, instead of traditional maths. It only ran for 5 years before they stopped it. When I left school and went to tech-college, I had to do 2 years extra trad maths courses to catch up. I didn't grasp modern maths as a pure subject because I couldn't see what use it was, or think of an application. This video apples that maths to examples, and for me gives it a reason. I wish they had done it as applied maths at school, rather than pure maths! 50 years after doing that school-work, the penny has dropped, and I understand. Too late for me, but knowing what I do now, I'd have asked for applied examples.
Thank you! The phase plane was taken from another website and I edited it onto the desmos graph actually. You can make whatever phase plane you want here aeb019.hosted.uark.edu/pplane.html For the graph at 17:30 I'll link that in the follow up video mentioned in the pinned comment.
Your videos are awesome, but you should never think why you are not getting too many views and subscribers, because there are just not two many people who wants to deeply understand math as well as know English, but who do watch are helped, develop interest in math, and do appreciate, so keep doing it, you are doing great job
Fun stuff! Another neat example is the compliance matrix in Mechanics of Materials. Here, the eigenvalues teach us something about how the elastic behaviour does its thing.
Here's another use of eigenanalysis that I came across recently: I wanted to figure out, on average, how many moves can be done on a rubik's cube from any given point, particularly for generating trees of possible positions. First, some ground rules: there's a total of 18 face turns: {U,D,R,L,F,B} × {90°, 180°, 270°}. We represent turning the top side 90° cw as U, 90° ccw as U', and 180° as U2. This is the same for the other five sides. Anyways, whenever you do one move, you don't want to turn the same side the next time. For example, R U U is the same as R U2. If we're generating a tree, this would mean we check both of those, and it would cost a lot of extra time. Even more so, R U U' = just R. Once again, we're undoing progress. Okay, so the first move there's 18 possibilities and after that there's always 15, right? Well almost. There's one other sneaky redundancy that we need to handle. Some moves commute, namely U&D, R&L, and F&B. So doing R L is the same as L R. And even more problematic, R L R = R2 L = L R2. Well to get around this, we simply say that (for the sake of generating) R turns can be followed by L turns, but L turns cannot be followed by R turns. With that, these two examples only have one valid representation, R L and R2 L. We do similarly for U/D and F/B. So now we're able to start calculating how many more states can be reached on any given layer from the previous one. We're not going to account for things like R U' R' U F R' = F' U F U' R' F or R2 U2 R2 U2 R2 U2 = U2 R2 U2 R2 U2 R2 as they are way too complicated, and it would be unrealistic to have the tree generator take the time to exclude those. So the first layer is just the solved cube. Next layer has 18 cubes, one for every turn. After that, it gets complicated. But if you were to express this as an adjacency matrix, this becomes much more reasonable. You could do it as an 18x18 matrix, I did it another way though; I did it as a 2x2. The first dimension is the number of cubes ending with U R or F, the second being ones that end with D L or B. Then the matrix simply states how many moves can be done on those sides after doing the previous one. For U R & F, you can do two of first group and all the of the second. For the others, it's just two each (Eg U can be followed by R F, D L B; D can be followed by R F, L B). So the matrix comes out to be [6,9;6,6] (each side has three turns). And the eigenvalue is 6+3√6≈13.35. This looks about right, some cubes have 15 moves they can do, others have 12. So now I can confidently say that each layer is asymptotically is 13.35 times bigger than the previous! Also, you could definitely do it with the 18x18 matrix, I just didn't feel like typing it all into a calculator and messing up some of the numbers. Anyways, I was then also able to get the qtm (quarter turn metric, same thing but only 90°turns) by saying there's basically a "first" and "second" cw turn for each side, ie R can be followed by R but only once. Reduced this becomes a 4x4 matrix, to which I have forgotten the eigenvalue, but it was around 9.5. At some point, I'll probably be able to get the eigenvalue for stm (slice turn metric) as well! (it has much more complicated restrictions)
this is a masterpiece may god bless you. I thought I understand these topics because they are too simple I uses them a lot but now my understanding deepened
Wow! An excellent instructional video. I covered this material over 25 years ago with a far weaker understanding given the quality of the teachers at the time. :)
I’m studying Dynamic Control Systems which involves differential equations just like the Spring and Damping scenario shown in video but at 2nd or 3rd order. And when our professor whips out some matrix stuff and put the coefficients inside really confused me but thanks for this video, I finally understand the reason behind that.
love it -- excellent explanations (just the right level/speed), and a fantastic compilation of the best examples of the use of eigenvectors/values -- thank you!!
17:19 You can see that it spirals into the origin because the absolute value of the eigenvalue is less than 1. E.g. imagine multiplying something by 0.99 over and over again - it'll eventually go to zero. (By absolute value of a complex number, I mean the length of the hypotenuse of the right angled triangle formed by the two coefficients)
You can find the vibration modes calculating the eigenvecs and eigenvals from a form of the rigidity matrix of an element in a structure, the element might be just a colum or even a full elevation.
This introduction to linear dynamical systems is great. A follow-up video on chaotic systems, their phase planes and dependence on initial values will be even better, since most engineering classes don't delve into that.
This is the first video of you I watch and I am a fan now. Thanks so much for the explanation. I would like if you add example of vibrating multidegree of freedom systems as we get eigenvalues and vectors for them too as I am sure your explanation will add too much.
I was studying inertia tensor which (I didn't knew what it was .)It was helpful for me . I love physics but maths is MY FIRST LOVE , AND I WILL NEVER LEAVE IT
This was quite interesting. I know all the tools described herein but I hadn’t imagined how they could be put together. This linear algebra isn’t just the tools, it’s they way you think about problems that it brings out. Very nice set of examples. You’ve brought out the multiple dimensions of problem applications. Might say you’ve shown us the fields that linear algebra applies to (pun intended).
Woww ,you are awesome... Thanks a ton for this wonderful video . I was only knowing mathematics but was unaware of its uses ..you made me understand the subject with purpose . Thanks again to decipher this important aspect of complex mathematical world.🙏
You touched on linear algebra, differential equations, and probability/statistics, three of the four areas of math that underlie most engineering and computing applications today (the fourth being number theory.) Nice video.
What about graph theory?
also zombies
Imgay Asheck that can somewhat fall under linear algebra since there’s matrix multiplication involved
I'm soooooo confused 🤪
@@imgayasheck595 Graph Theory is more or less a part of Linear Algebra, since a graph can be represented as a matrix
You and grant Sanderson are the reasons my interest in "understanding" mathematics is still alive ... I just can't express in words how much I owe to you both 🙏🙏🙏huge respect from India
Same
You should also know Mathologer.
How's the progress going on, Shama ji?
It's a bit sad because you need someone (other than yourself) to cultivate curiosity towards math
This was a better explanation in 20 minutes than anything in my Diff Eq + Lin Alg class. I just finished a test on eigenmagic with a decent grade, and I didn't know what an eigenvalue even was.
Great quote from the video, “Eigenvectors/Eigenvalues (in many cases) reveal long term behavior.” Why is this not the leading sentence for every math class discussing this topic? All of the sudden its application became extremely clear.
Having watched all the previous videos, this one felt a lot like a sequel and I simply loved it. As a Math teacher having to teach Linear Algebra to engineering students, your videos are a great source of inspiration! Thank you!
Aww yes ! That feeling when you just finished your masters in computer science and everything on this video made perfect sense.
sana ol nakapag masteral
🤩 I just got my Masters. 😂
@@jj74qformerlyjailbreak3 nice
@@cruzmarco6048 Nothing but ❤️ my friend. 👍
Chi chi would be proud
Dawn, you can turn yourself into an engineer just by watching this and other few channels on youtube (believe me, I'm an engineer). It's so well explained and easy to understand that makes real classes tiring and boring. Thank you so much for spending your time making those helpful videos!
I don't know bro, you gotta do a lot of exercises haha
I am not in college, but videos like this is rekindling my interest in mathematics, which I love more than crosswords.
@@BooyahLThat's true, but most of the concepts you can find here on youtube, even some applications/projects and advanced exercises. I know it doesn't replace a real engineering course, but there are so many things that you find here on youtube that if you had a guide of what to study next, you would have a pretty good idea of what is a engineering course.
@@RM-zx9ee And if you just start zooming around Wikipedia pages on math, you could learn a lot... more than you could possible absorb all of.
I used to think that too but lately I’ve changed my stance. Learning is iterative and I think what makes videos like these so good is normally because this isn’t your first pass over the subject
No joke, your content has singlehandedly revived my interest and passion for pursuing both learning and teaching math and algorithms. I’m going to really, finally, put my nose to the grindstone to cop an ICPC medal off this inspiration high. You’re the best, Zach, and you have a new Paterson member. Godspeed!
Dude what a donation! Seriously appreciate it and hope you go pursue those interests and passions, best of luck!
So, I'm from Uruguay (Our native language is Spanish), and I was hearing the video (Started with a related one, of the same channel) and thought: "Ok, eigenvalues, and eigenvectors are really cool! So why I'm not learning this at University?", And I searched the translation, that is: "Valores propios, y Vectores propios" respectively.
I'm shock.
I have been calculating the eigenvalues, and the dimensions of the subspaces associated, all over the place, but no one has ever teached us what the hack they are used for!
Yeah... I'm probably dumb because I didn't searched about this before, and, I know, some times is hard to teach "all the aplications" of an unknown tool, (for the studient). But, is just far more engaging and useful to understand why i'm using "det(A-xI)" and solving a weird polinomium.
Anyways. Thank you SO much, for the video.
Autovectores y autovalores se suele usar en los libros de texto en español también
This is very true. In physics, and especially in Quantum Mechanics we use eigenvectors/values all the time. It's really fundamental - but my course (many, many years ago) never once covered what they are geometrically or even in some depth mathematically. They were simply presented as a series of algebraic manipulations of arrays of numbers to get 'the answer'.
To this day, even though I can calculate the probability amplitude of spin states of electrons (for example) I haven't the faintest idea why this model works as it does and how it came to be invented in the first place.
Hope you guys enjoy! Two things I want to mention
1) This video (mostly towards the end) references some things I discuss in the applications of matrices video ( ua-cam.com/video/rowWM-MijXU/v-deo.html ). If you haven't seen it you'll be fine for this video, but a few things will make more sense if you have seen it.
2) Here's the video where I link and explain the desmos file I used for this one. ua-cam.com/video/1_fXgJWQhVY/v-deo.html
like could you please make a video on desmos for animation and stuffs...it would be of great help!!!!
Make a vídeo in control theory plz haha
Genius in Artificial Intelligence
I'd love to see a quantum mechanics application of eigenvectors.
Love your videos Bro.. in an age where school is killing interest in the sciences, your videos keep my curiosity and interest alive... ❤️❤️
Sukalyan Roy truf
School isn't killing interest in sciences. It just fails to keep up/compete with the insane amount of interactive, appealing and sometimes addictive content available online.
Daily life is THE perpetual school of sciences, be it material science (build a house, make a dress), math science (budget, electricity), language science (talk to your neighbor of dissimilar roots) or history science (voting), and only peer pressure is killing interest in being interested, therefore you, alone, are responsible for keeping your interests alive. I'm sure you will go far, maybe one day you will be in a position to direct the interests of other, to catch someone who is falling to peer pressure and show them the benefit of joining you in the journey.
@@b_kind force-feeding subjects instead of giving insights about applications is a frequently complained part of schools though, even on subjects that are very useful. That probably contributed to the "killing"
You said it, I can say first-hand that my highschool is ruining science and especially math for the majority of students, it's truly a shame
I had the pleasure of being directed to your videos today. Thank you tech companies (who paid attention in Linear Algebra classes) for using what I suspect to be a low-rank matrix decomposition to learn about my viewing preferences.
When the eigenvalue is imaginary, that means you got out of R² , and the eigenvector is perpendicular to the plan, that is the 3D vector [ 0 0 1 ], it's actually the rotation axes for the 2D rotation, that's our eigenvector in that case..., this also happens when solving vectors in 1D and getting the value “i” as result, that's actually the vector [ 0 1] in 2D => the vector perpendicular to the subspace we are in
(+newdhia hassen) Got it. Kind of like curl.
Thanks, that was helpful. :)
I was thinking in similar lines since the imaginary plane is just the other plane of physical value that affects the current observed value we are dealing with. it's just in a different plane. Nice one.
oooh
That's a nice interpretation.
This is great! Thank you! Glad to be your Patreon supporter and to be able to see your videos first!
Thanks for the continued support Nick! Glad to keep seeing you in the comments
Thanks you very much, this completely boosted my understanding after studying this a year ago and not knowing any practical applications or even the essence of studying and understanding them...i'm grateful for time and effort you took to make such a clear video...
Been binge-watching your transformation video. Although i've done my signal processing course (which i suck at), your video gave me new insights on how to look at those transformations, big thanks! :D
Dec 12:
Me: finishes finals for differential equations and linear algebra.
Dec 13.
Major Prep: Eigenstuff
Me: well, the timing couldn't have been better
Since we talked about the google page rank algorithm it seems fit to tell you that your exam probably caused you to be here :)
Wow, this was huge. The simple explanation you started with got me to get all excited every time I saw the eigenvectors coming up! Really nicely done!
When I got to the part about stability and rotation I was like: “Hey Laplace Transform!”. I love the way so much of math overlaps!
Feedback: The last videos have been awesome! They speak to the intuition, and allow people a much better understanding of how things work and when they apply. You've done a fantastic job - thank you for doing it. This is the kind of video people should keep returning to until the info it contains stick - it is integral to understanding pretty much any complex system. This really helps! 👍
This was really helpful. I'm watching several videos on eigenvectors and eigenvalues right now because I did have this stuff in college, but back then, it wasn't well explained as it is with the animations you roll back and forth. I really appreciate this. Thank you.
The fibonocci series example has solidified this concept in ways that I've been searching for
We spent like 2 weeks learning about these in my math paper, I had no idea what any of it was about, but at least I could do the arithmetic. This video's visual explanation is soo much better for understanding what these are than any math text book
watching this now after just finishing my bachelors in mechanical engineering and going onwards to a masters in computational engineering. feels like the most amazing application-based review video. THUMBS UP!
4:50 note: to find the eigenvalue (1+sqrt5)/2, you use the equation (1-λ)(0-λ)-(1)(1)=0. to get this you set the right side equal to λ(F_(n-1) F_(n-2)) where λ is the eigenvalue, then get everything to one side. now you have a homogeneous system of equations. the matrix of coefficients for any homogeneous system has a zero determinant due to the right column of zeroes. use this to set up a quadratic equation and solve for λ using the quadratic equation
I understood the geometric interpretation of eigenvalues/vectors pretty well, and had a working knowledge of the applications in DEs and things like PCA, but without a real intuition for why it worked - this is the first video I've seen that really connected them for me, thanks.
I've been an engineering student in various forms for 5 years and I've studied the mechanics of calculating Eigenvectors and associated values for hour and hours. I've searched and searched for a 'simple' explanation of what an Eigenvector actually is. This is by far the best explanation I've found.
I had a brilliant doctoral computer science professor this semester and he pretty much told us that physicists and engineers he collaborated with over his career most of the time wanted fast ways to solve for eigenvalues on large computer systems lol
Who else saw black spots on the orange dots around 11:54 (arbitrary time of course) that disappeared when you looked directly at them?
would.'ve really been helpful for when i was in college. profs just threw these things at us, i had no clue till this day.
I just took my linear algebra final, and this video answers the question my professor failed to all semester: what is the point of all these techniques I'm learning?
So true. I feel the exact same way. I wish math classes would include more material on application.
Why are you studying maths if this is a question you care about?
@@SpaghettiToaster I'm a computer science major, and this is a requirement. If I wanted to study something for it's inherent beauty, I'm be a liberal arts major.
@The Thunderblade Damn dude, I was joking, but where at all did I say say anything approaching "what is maths?"
Regardless of the subject, a student's going to grasp the material better with some understanding of context. For example, Mondrian's famous abstract art seems pretty simple if you see it out of context (you know, the ones with the thick black lines and geometric shapes in primary colors), but when you appreciate what the guy was able to do earlier in his career, and why he chose to go abstract instead, it lends a better understanding of why his art is so impactful.
Check it out, I have no problem with learning advanced maths. My issue arises when I, along with a good chunk of my class, were asking in good faith, "what do these numbers we're coming up with mean?" I get that there are people who enjoy doing math just to see the formulae churn out answers, as that's who my LA professor was, but it's not me. This guy was teaching at a superficial level of just showing us various techniques without bothering to give us any insight into the larger implications of what the math meant, and it made the course more of a headache than it needed to be.
Maybe go talk to someone about unleashing that kind of hostility on a total stranger, though. Can't be good for you to carry around that level of vitriol.
@@SpaghettiToaster I thought the question quite valid and mirrored my own when reflecting upon how and why the curriculum I followed was created. There must have been a perceived context within which what was to be taught was decided. As an example, although I did Linear Algebra we didn't touch upon Eigen 'stuff', however in a Numerical Analysis we were shown how to calculate the eigenvalues of n-dimensional matrices. I did wonder why. It was only after I graduated and taught myself Quantum Mechanics, starting with The Copenhagen interpretation that I came across Eigen 'stuff' and the penny dropped as to why. Mathematics in and of itself is beautiful but in a course created by a random subset of planetary mathematicians it can be useful to know the why because that hints towards the collective understanding of the Department's research staff and 'the arrow' of their vision. One always be brave enough to ask 'why' and not just settle for being told.
I cannot thank enough for this video. I never quite understood what eigen values and eigen vectors are until I saw this video.
I don’t know what eigenvalues and eigenvectors are essentially until I watch your great video. Thank you!
Thanks man. You have my respect. I have taken courses on linear algebra in my college, but the way you explain won my heart. This motivates me to learn about it more, whereas the studies in college were mostly numerical and boring. Thanks for all knowledge you are contributing to this world! Keep making such videos. I love your channel.
This Channel has it all I need, comedy, academics, and a Star
Wow! I'm a machine learning engineer, and I have had studied all these concepts time and time before in the college, but I had never grasped the gist of it this good, really surprising! thanks!
I am always waiting for your videos.
I feels like 20 min of your video teaches more than my whole 8 hours of collage .
Big thanks.
💖
My brother I am from Nigeria's fall system of education so I understand . You don't just feel...you KNOW!
Great video ... one of the gold nuggets of UA-cam ... first time I saw some real applications (just was number crunching to me before this). Thanks ... it is most appreciated!
This is an excellent demonstration. I've been struggling with the concept of the Eigenvector for some time and have explored many sources for different explanations. This is by far the best, I think I may almost understand their purpose now. Please keep up the good work!
I feel every engineering school should show these videos to show the reason why they are learning what they are
Or maybe they could be able to inspire their students without the help of a youtuber, on account of being salaried professionals?
Thanks!
Fun fact for the mathematically inclined: This demonstrates a nice transition of 'Pure Math' theory followed by its 'Applied (Math)' to real world problems &/or situations....it's ALL about PRACTICE & MORE PRACTICE friends! Superb tutorial btw!
As a new student who can barely factor polynomials, this is a fascinating look into a topic I don't understand and can't perform yet. Thanks for making it digestible for those of us in our early struggles. These videos make me want to push harder and get myself to the point where I can understand and apply the concepts.
Never seen a more intuitive video about eigen vectors than this!
i appreciate this video. seeing real-world applications by using abstract math is one of the most profound things i have witnessed. knowledge is the real power behind being effective in the world. thank you for educating us. it makes the world a better place. (and a less ignorant place)
Dude. This video is excellent. Finally getting more intuition on why this eigenstuff is so useful. Thank you.
Glad you liked it!
An early Christmas present. Thank you sir!!!
Your video is pure joy for me as a math lover.
Im an engineering student( graduate level) with specialisation in control systems theory.
I've seen many explanation videos, but yours is my favourite.
If your video had been online while i took my math courses in undergraduate level, i could have saved a lot of hours :D.
You have done a great job!
These are the videos that make UA-cam great! Please continue to make this type of videos, I absolutely love them
As a postgraduate student in engineering, I sincerely recommend this video to all engineering students as most of our modules make use of matrix and eigenvalues! And this video really answers the question that I always have in mind "why characteristic polynomial and eigenvalue are so important"
you explained years of trying to understand linear algebra in one split second. Thank you!
YOU ARE AWESOME. WORLD REALLY NEED TEACHER LIKE YOU. YOUR INFORMATION ARE WAY MOREEEE USEFUL.
THANKYOU
Man this is brilliant. I saw this video about 4 months ago. Lately been going through matrix n clustering in python and it hit me and I rushed back to this vid. Everything makes perfect sense now. Am happy it clicked 🚀
This video is mindbogglingly good. Outstanding!!! You will have a new Patreon supporter shortly. I might have taken a different career path if I had watched these videos earlier in life. Outstanding.
Thanks so much! Hope the channel continues to be helpful.
love these videos you've made on linear alg. Im a soph ME student finishing up a course on lin alg and I've found your videos really entertaining, clarifying and well produced. Ever considered doing a video on basis vectors? properties and apps of the determinant? This is such useful math and I just wish my teachers could make these abstract topics as palatable and engaging as you have.
Your videos are really helpfull as they graphicly and schematicly describe mathematic and physic concept that other way would be almost imposible to understand, thanks Mr. Zach for the time and effort you put on making and posting your videos on UA-cam for the public. Me and lots of other people really appreciate it.
the best video on eingen-stuff i've ever watched! thanks a lot. i've been searching for this type of explanation for a long time
Someone may have already mentioned this, but the idea of transitions and equilibrium ties directly to the concept of limiting probabilities in a system - particularly in Markov Chains, under the right conditions. I find the connection to eigenvalues and eigenvectors in this context fascinating, and I really regret that we didn’t explore this aspect in my university course on probability and stochastic processes!
You and 3B1B are killing it. Linear algebra is the most abused subject for whatever reason by professors. Great vid.
Zach, this is a great video, just in time to my exams, now i unsderstand the perpose and how much usefull are eigenvectors and eigenvalues, colleges dont teach math in the way they should, they dont explain "why is that important?" they just give u a book. U can find motivation watching this kind of videos, congrats!
One of the best video on understanding Eigen value problem in practical way..Thank you..
Thank you so much, your explanation is simple and clear and contains alot of concepts which makes the video not boring(which math videos usually are) and intriguing
Good video, thank-you. As a child I was part of a pilot experiment that taught 'modern maths', (SMP 1-5), from 11-17 yrs old, instead of traditional maths. It only ran for 5 years before they stopped it. When I left school and went to tech-college, I had to do 2 years extra trad maths courses to catch up. I didn't grasp modern maths as a pure subject because I couldn't see what use it was, or think of an application.
This video apples that maths to examples, and for me gives it a reason. I wish they had done it as applied maths at school, rather than pure maths! 50 years after doing that school-work, the penny has dropped, and I understand. Too late for me, but knowing what I do now, I'd have asked for applied examples.
Superb, right up there with 3Blue1Brown imo......Bud, can you share some of those graphs ? especially the one's @ 11:00 & 17:30
Thank you! The phase plane was taken from another website and I edited it onto the desmos graph actually. You can make whatever phase plane you want here aeb019.hosted.uark.edu/pplane.html
For the graph at 17:30 I'll link that in the follow up video mentioned in the pinned comment.
Damn, I didn't know this when we learned this. They should have told us this first.
you weren't ready. And you would have complained anyway.
@@donegal79 Aren't we all not ready for what school is going to teach us? Should we go through the entire textbook before attending the first lesson?
Your videos are awesome, but you should never think why you are not getting too many views and subscribers, because there are just not two many people who wants to deeply understand math as well as know English, but who do watch are helped, develop interest in math, and do appreciate, so keep doing it, you are doing great job
Are you freaking kidding me? Now it makes sense, i needed this in college :( Thank you man it was fun watching this, it clicked!
Wow, thus series of applications.. Is the best thing you are doing.. And the best thing going on the internet
Fun stuff! Another neat example is the compliance matrix in Mechanics of Materials. Here, the eigenvalues teach us something about how the elastic behaviour does its thing.
Here's another use of eigenanalysis that I came across recently: I wanted to figure out, on average, how many moves can be done on a rubik's cube from any given point, particularly for generating trees of possible positions. First, some ground rules: there's a total of 18 face turns: {U,D,R,L,F,B} × {90°, 180°, 270°}. We represent turning the top side 90° cw as U, 90° ccw as U', and 180° as U2. This is the same for the other five sides. Anyways, whenever you do one move, you don't want to turn the same side the next time. For example, R U U is the same as R U2. If we're generating a tree, this would mean we check both of those, and it would cost a lot of extra time. Even more so, R U U' = just R. Once again, we're undoing progress.
Okay, so the first move there's 18 possibilities and after that there's always 15, right? Well almost. There's one other sneaky redundancy that we need to handle. Some moves commute, namely U&D, R&L, and F&B. So doing R L is the same as L R. And even more problematic, R L R = R2 L = L R2. Well to get around this, we simply say that (for the sake of generating) R turns can be followed by L turns, but L turns cannot be followed by R turns. With that, these two examples only have one valid representation, R L and R2 L. We do similarly for U/D and F/B.
So now we're able to start calculating how many more states can be reached on any given layer from the previous one. We're not going to account for things like R U' R' U F R' = F' U F U' R' F or R2 U2 R2 U2 R2 U2 = U2 R2 U2 R2 U2 R2 as they are way too complicated, and it would be unrealistic to have the tree generator take the time to exclude those. So the first layer is just the solved cube. Next layer has 18 cubes, one for every turn. After that, it gets complicated. But if you were to express this as an adjacency matrix, this becomes much more reasonable.
You could do it as an 18x18 matrix, I did it another way though; I did it as a 2x2. The first dimension is the number of cubes ending with U R or F, the second being ones that end with D L or B. Then the matrix simply states how many moves can be done on those sides after doing the previous one. For U R & F, you can do two of first group and all the of the second. For the others, it's just two each (Eg U can be followed by R F, D L B; D can be followed by R F, L B). So the matrix comes out to be [6,9;6,6] (each side has three turns). And the eigenvalue is 6+3√6≈13.35. This looks about right, some cubes have 15 moves they can do, others have 12. So now I can confidently say that each layer is asymptotically is 13.35 times bigger than the previous! Also, you could definitely do it with the 18x18 matrix, I just didn't feel like typing it all into a calculator and messing up some of the numbers.
Anyways, I was then also able to get the qtm (quarter turn metric, same thing but only 90°turns) by saying there's basically a "first" and "second" cw turn for each side, ie R can be followed by R but only once. Reduced this becomes a 4x4 matrix, to which I have forgotten the eigenvalue, but it was around 9.5. At some point, I'll probably be able to get the eigenvalue for stm (slice turn metric) as well! (it has much more complicated restrictions)
You're an absolute psychopath, bro🔥
@@leoe.5046 ah thanks man
No idea why I needed _that_ many words though...
this is a masterpiece may god bless you.
I thought I understand these topics because they are too simple I uses them a lot but now my understanding deepened
Wow! An excellent instructional video. I covered this material over 25 years ago with a far weaker understanding given the quality of the teachers at the time. :)
Fantastic. This plus 3B1B gives a really good understanding. This video was really awesome bro.
Very well done!! Great work. When I was learning these in linear algebra, I had no idea what they were for.
I’m studying Dynamic Control Systems which involves differential equations just like the Spring and Damping scenario shown in video but at 2nd or 3rd order. And when our professor whips out some matrix stuff and put the coefficients inside really confused me but thanks for this video, I finally understand the reason behind that.
love it -- excellent explanations (just the right level/speed), and a fantastic compilation of the best examples of the use of eigenvectors/values -- thank you!!
17:19 You can see that it spirals into the origin because the absolute value of the eigenvalue is less than 1. E.g. imagine multiplying something by 0.99 over and over again - it'll eventually go to zero.
(By absolute value of a complex number, I mean the length of the hypotenuse of the right angled triangle formed by the two coefficients)
One of the best videos that I have found on this topic! Great work!
Super video! I applauded for $2.00 👏
How did you read my mind ?? I actually needed this video😀😀 I love this content ...this is very unique
This was literally so beautiful!
Very nice video. An extremely efficient way of visualizing and teaching math and engineering concepts. Thank you.
Thank you. This make the whole thing more intuitively understandable. Love your vids a lot
thanks for everything u do man. its just shows how simple maths can solve some of the greatest questions of humanity
You can find the vibration modes calculating the eigenvecs and eigenvals from a form of the rigidity matrix of an element in a structure, the element might be just a colum or even a full elevation.
Just wow! A great visual way of explaining the fundamental concept in Linear algebra. Thank you!
I might not understand all the maths,
but I won't stop me from being mesmerized by all of it
This introduction to linear dynamical systems is great. A follow-up video on chaotic systems, their phase planes and dependence on initial values will be even better, since most engineering classes don't delve into that.
This was one of the best videos I have seen explain system stability
I really like this video because now I feel like I actually understand something I rotely memorized in class. Thanks!
This is the first video of you I watch and I am a fan now. Thanks so much for the explanation. I would like if you add example of vibrating multidegree of freedom systems as we get eigenvalues and vectors for them too as I am sure your explanation will add too much.
This was very very illuminating. You’re an excellent teacher.
I was studying inertia tensor which (I didn't knew what it was .)It was helpful for me . I love physics but maths is MY FIRST LOVE , AND I WILL NEVER LEAVE IT
Brother , you are doing a great job !
Hope we get more interesting videos on some weird topics
Although I don’t fully understand, I’ve been amazed! Such a brilliant video! Thank you very much!
Thank you for this video. I've always wanted to understand what eigen stuff are ever since hearing about them in college.
This was quite interesting. I know all the tools described herein but I hadn’t imagined how they could be put together. This linear algebra isn’t just the tools, it’s they way you think about problems that it brings out. Very nice set of examples. You’ve brought out the multiple dimensions of problem applications. Might say you’ve shown us the fields that linear algebra applies to (pun intended).
This is very well put together and even greater visualization than I got in Uni!
This was a golden video, straight into my favourites. Thanks!
Woww ,you are awesome... Thanks a ton for this wonderful video . I was only knowing mathematics but was unaware of its uses ..you made me understand the subject with purpose . Thanks again to decipher this important aspect of complex mathematical world.🙏