A quick trick for computing eigenvalues | Chapter 15, Essence of linear algebra
How to write the eigenvalues of a 2x2 matrix just by looking at it.
Need a refresher on eigenvalues? • Eigenvectors and eigen...
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Lockdown math lecture talking about the mean product formula:
• The simpler quadratic ...
Timestamps:
0:00 - Background
4:53 - Examples
10:24 - Relation to the characteristic polynomial
12:00 - Last thoughts
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Maybe not as visually focused as other topics on the channel, but this came up while I was doing the next installment for differential equations, so I thought it'd be nice to add a quick new chapter to the linear algebra series. I hope a few students out there find this helpful!
Thank you for everything boss. ❤️
Great work
All the visuals probably take you awhile. I think folks are happy with the short and sweet stuff occasionally. The content is already so high quality.
Once you get the eigen value’s how you find the eigen vectors.
@@aslpuppy1026 haha on Patreon he was debating adding that. I believe it’s coming in another vid.
When the world needed him most, he returned.
He vanished
3Blue1Brown: The Last Number Bender
like in the midst of my IB exams XDXD
@@justinyang21114798 Good Luck,
You got my 666 like. So do you believe in superstitions lol 😁
One moment of silence for every student that uses this method and doesn't get full marks because the teacher wants them to "show their work".
😂😂😂
Unless the question is 'write down the eigenvalues ' . Yeah, kind of renders it useless for most of us unfortunately 😅 maybe can use it to check your answers
Happened to me 😢
One thing you have to ask yourself is have you understood what eigen values are? Your teacher probably wants you to understand eigen values rather than deriving answers. The creator took 13 minutes and showed each step and work. If you only write the answer there is no way for teacher to know if you understood the fundamentals. Probably your teacher is grading your knowledge
The fact that he kept playing the audio really sealed the deal for me
Says a lot about the mechanics of learning!
Especially when we see it at the same time
I kept laughing everytime I heard the jingle, and the little 'p' was so funny at 6:08. I'm never forgetting this!
*ding*
Sorry but that "song" was annoying as fk. Overall good video though, as always.
@@TheRealFFS I highly disagree :D As a piano player, I found it superb and it entertained me a lot :D So catchy melody
Im listening to this at 1.5x which makes it more hilarious
@@TheRealFFS nah
Your "one sentence reminder" of what an eigenvector is was better than what I was taught at University. A student was shouted at by the professor when he asked if the prof could visualize it. The prof shouted: "We are an elite University! We don´t paint pictures, we abstract!" Yeah thank you - not MF. I´m happy there are people like you on KZhead!
Visualization is the art of abstraction.
weird, I've never met a mathematician who doesn't like to draw pictures
@@Czeckie Neither have I
@@Czeckie Yeah indeed. I often find visual explanations give the best intuition while the alegrabic is best for computation
@@Czeckie My dad had a quantum physics professor who thought visualizations only lead to misconceptions. I only know it because my dad says he learned nothing in his class, even though he probably made a decent grade, and always wondered what he was actually doing in that class.
8:11 for anyone wondering, this is the Stern and Gerlach experiment and it is absolutely fascinating. Look it up
He began the suffering of every physics students
Thanks, I like learning things
I'm gonna have that god damn jingle stuck in my head for days now.
11:19 if you want to hear it without moving the cursor
its my new ringtone
m ± √(m² - p) Ping!
10 hour loop when?
@@damiandeza2761 4:47 is a better timestamp.
Totally agree, this is such a nice way to compute eigenvalues. I hadn't explicitly thought about the connection to factoring quadratics before as you did at the end, very nice:)
Prof treffor. What a crossover here,
Both Trefor Bazett and Kyle Broder in the comments section, it must be a good video
I think there's a pretty wide ranging connection there (between matrices and polynomials), which is used all over the place in certain branches of mathematics, but rarely shown early on
Agree - I actually learned this trick through geometry and ellipses (or equivalently any conic section if you throw an i in there). The determinant term is your a^2 + b^2 terms (related to variance if you're dealing with stats) and the trace is the of sum of your major and and minor axes (I'm probably missing factors of 2 and 4 here). I believe the final formula looks directly analogous to the formulas for eccentricity and the semi-latus rectum and the only reason I know this because I was doing some work on physical mechanics and I re-remembered the connection between rotation matrices, orbital mechanics and conic sections.
My two favorite teachers!!!
The jingle is extraordinarily genius. First time I heard it, it caught me off-guard with its brilliance. Thanks to you both!
It was so good. I gave such a vague idea to Tim, and he came back with something hilariously catchy.
@@3blue1brown Props.
@@3blue1brown Do you think you can find a formula like this only for polynomials of degree 3 and 4?
@@3blue1brown the *ping* is what gets me
I love your channel. I studied Physics/Mathematics in college before calculators (Took tests with a slide rule, which I learned in elementary school), back in the early 70's. So I learned everything by studying formulas. Watching the newer generation of mathematicians/scientists use computers to visualize math is something we older math geeks could only dream of. It has really opened up the field to people that would otherwise be afraid to tackle the subject. I don't do math much anymore having been retired for almost a decade but I love watching these (and other videos) to see what the field is doing. Thank you so much...you bring smiles to us senior citizens.
I didn't use slide rules, we had calculators. But I also memorized the formulas and was an "A" student which just meant I memorized the formulas but until this series I never understood it. Now it seems to amazing and simple.
Irony is for most maths in university, a calculator simply doesn't help you (especially in the super pure fields such as analysis)
I'll point my students to this, I've tried showing tricks like this before in tutorials, but students often need reminders before it becomes a technique they can incorporate. This, as always, will be a great resource for them! 🙌
Wow! Both Kyle and Trefor in the comments section
maybe capture them and force them to listen to the jingle for a day
@@theblinkingbrownie4654 Whatever it takes 😂
I can’t believe that this particular video comes right along with my LinAlg Final!! Edit: i survived!
Same. My exam is litteraly in 2 weeks
Uni?
damn this came literally hours after my lin alg exam 😭
@@joseffnic3560 same, linalg exam today xD how did it go for u?
I literally just took mine like 2 hours ago
Germans: Hey, we came up with a nice term for these mathematical objects! English: lemme just translate the second half
What for Example?
@@spideybot5754 Eigenvalues for example.
@@w3lt3nbr4nd2 Thanks...
"Self vector" has not same feel to it...
@@tetraedri_1834 I would rather translate it as "own vector". "Self vector" sounds more like a translation of "Selbstvektor".
Ah thanks, we bigfoots are always finding ourselves needing to calculate eigenvalues out here in the woods
hi
Username checks out
Does your foot get even bigger sometimes ? if yes by what eigenvalue ?
Then you should use matlab bro
The fact that this video came right before my finals on 17th May is just testament to the fact that this man is omnipresent and knows exactly what his viewers need The best Linear Algebra series on the internet, without a shadow of a doubt
Everytime I see that blue circle in the notifications I genuinely smile and I thank God I get to live in the same times as this amazing teacher
I just wanted to say I saw a dog one day and she had blue eyes but one of them had a brown section (¼ of the eye), so I mentioned your youtube channel to the owner and how the dog has a 3blue1brown eye, and the person actually knew about your channel already. It was a great moment :)
This channel is one of the most inspiring math channels Ive found and is largely responsible for how well Ive been doing in my college math classes. Bless 3blue1brown
IDK how this is in US, UK and EU but in post USSR countries we learn this trick to get the roots of quadratic equations in school alongside the standard quadratic formula. At least very similar trick which I think is totally equivalent. IIRC it's called Vieta's formulas. (Sorry for bad English)
Yup, in Vietnamese education system, its called "Vi-et" theorem, probably borrowed from USSR guys
NEW LINEAR ALGEBRA VID DROP LETS GET EM
I love how he just comes out casually with a masterpiece every single time
One of the best mathematics related videos I've ever seen. They way you explore the intricacies of the calculations and then relate them to real world problems with insightful visualizations inspires me to learn more math, computer science, and physics. Thank you so much.
I'm learning data analysis and a few days ago went over the Principal Component Analysis algorithm. Your videos and also the playlist of Victor Lavrenko were super helpful. I'm glad to see that you still add videos to this playlist!
This is amazing! Too bad I'm now doing my bachelors thesis - would have helped me a hell of a lot for my first to third semester! Thank you very much for your amazing content!
As an engineer who had multiple classes that dealt with eigenvalues and eigenvectors, I was always delighted to discover more ingenious ways of calculating them. Mohr's circle has got to be one of my favorite methods.
There is also a variant for Mohr's circle that applies to 3x3 matrices, and using another geometric trick for solving cubic polynomials, I've found it fairly easy to calculate things quickly and efficiently.
Hey, Grant, I just want to say thank you. I used to be terrified of the idea of taking calculus; however, your essence of calculus series really gave me insight that makes at least the basics make intuitive sense. I’ve since became very interested in maths, and I wouldn’t be there without you
I have just found this channel through The essence of calculus series and I want to congratulate you for how good you are making these videos. Thank you.
3b1b videos are so good, I give them a like before they even start playing.
I absolutely hate these kinds of math, but since my studies recquire them, all these videos are a godsend. Thanks a lot, and please keep on making more so students like me can bear all these abstract concepts !
i recently came back to this channel after recommending a friend check out your "lockdown math" series for a brief but thought-provoking review of some high school math topics. i love that we have a song for the simpler quadratic formula now, especially after hearing you say it was too short for a song. this video made me smile, thanks :)
Your YT channel is a treasure, a feast for mind. Big thank You to share this materials with us.
I'm not sure I'll ever have to calculate eigenvalues, but I sure as hell won't be getting that jingle out of my head now.
studying a machine learning master and being able to relate to this guy is the best feeling in the world. thank you 3B1B
Back in high school I was really good at math. I remember studying my father's university math text books when I was bored, and I did competitions and all that. However, at university I weirdly chose for something that doesn't have anything to do with math. This channel keeps my interest alive, I appreciate it!
I just learned about the characteristic polynomial last week, so this trick is golden! Thank you!
Nice application of the trick usually associated to Po-Shen Loh
Came to the comments to find this! I guess I shouldn't be surprised that a "trick" for solving quadratic equations would apply to finding eigenvalues of a 2x2 matrix! (although I have to hand it to Grant - combining it with the fact about traces and determinants was a beautiful addition)
Yes. I think Po-Shen Lo should be mentioned and credited!
MY FAVOURITE SERIES IS STILL GOING?!
This is eerily specific timing right as I'm learning about eigenvalues in linear algebra and differential, so I'm excited to try it out!
I watched this a week ago, and I use it almost everyday now! Your videos are amazing.
I've been following this channel since the Euler's characteristic formula video. I remember when I saw a couple of years ago the first linear algebra series videos when they were being released. I was in high school still. And this channel showed me a first glimpse of what real mathematics is like. It seemed so advanced to me. I had never seen a matrix nor linear transformations, so it was surreal to me. With those videos I for the first time felt I deeply understood something in math. Before that, it was a lot of memorization. And it really changed my perspective and my way of doing things. (Maybe I got a bit too fanatic at that point of trying to understand deeply absolutely everything, and I was a very slow student because of that.) Now, I see these videos and they seem so basic to me. I knew what was gonna happen at pretty much every point in the video, so it was kinda boring. I feel so sad that it is this way. It's also great because it means I learnt a lot. I do feel very nostalgic at hearing this music too though. God, I feel like an old man saying all of this.... Cheers everybody! Math is beautiful and this channel too!
I dropped out education after my master's degree. because all that we were doing was based on memorization with out knowing what they really are. now after watching 3 blue 1 brown intuitive lectures I have taken admission for higher studies ( Ph.D in maths)
Don't do a PhD just because of some KZhead videos, you will burn out
Don’t you have to complete your master’s degree in order to do Ph.D ?
@@Caleepo in most countries, yes, but there are exceptions where one can do their phd with only a bachelor degree (like the UK)
@@LiloudOr oh ok, thats weird
@@Caleepo From what I was told is that in the US, many people don't do a Masters and go straight into a phd after their bachelors, but the first 2 years of such a phd are like a masters where you still have to take lectures and seminars, and the phd can thus take more than 5 years to finish. Not sure if it counts as if having finished a Masters if you decide to drop out after 2 years, thus I don't really see the point of it. Perhaps it's about getting phd scholarships which pay better and you have a better chance to get than a scholarship for a masters degree and colleges in the US are super expensive.
Our professor just mentioned this at a side note but I'm so happy to have this clear, self-evident proof and visualization. These videos are always a blast, I can't ever thank you enough for all your great work. Without your videos I maybe wouldn't study engineering science today :) Greets from Berlin, keep it up!
You're one of my hero's. I actually have returned to Uni for a second degree majoring in math. I am now fluent in the languages of diffusion and waves and so much more. Best decision of my life and you are a definite contributor. Thank you.
Was not expecting another entry to the linear algebra series! This series helped me gain so much more intuition and appreciation for linear algebra when I would've been clueless and spiteful towards the subject otherwise. Grant, if you're reading this, a video or even a whole series on complex analysis and complex derivatives/integrals would be awesome. I'll be taking a class like this over the summer and I feel like there is lot of visual intuition to be gained underneath all the equations (like most of math it seems)
I was literally just rewatching your Linear Algebra playlist
I'm in college (aerospace engineering), I literally have a test on eigenvalues and eigenvectors TOMORROW, you've saved my life dude
I got so proud seeing the "p-q-formula" in the spotlight! It is the way we're taught to solve quadratic equations in Sweden.
Nooo your eigenvalue video came out the day after my linear algebra final. At least the other videos could help
Well when I studied linear algebra came up in a bunch of later courses, even eigenvalues.
Awesome that Acapella Science is getting some of the recognition it deserves!
I love the calm, classical music accompanying these videos.
Thanks so much for your content! I took a coding class this semester and was able to make a 2.5D game with isometric gridding in Python, and it's because you taught me how to think of matrices as transformations of space. It was really awesome to see how consistent what I learned was, you're incredible! God bless!
I love the acappellascience jingle! I honestly wouldn’t mind if you got him on board to sing equations every video, adding that extra sonic dimension would complement your amazing visuals and aid in memorization as well :)
Coincidentally, I have my linear algebra final tomorrow. Thanks for this, Grant!
ur a hero for this, as we’re about to dive into systems of differential equations in my ODE class
It was quite surprising to stumble upon a Grant's video on eigenvalues in recommendations this morning, as only yesterday I was trying to recall the general procedure of calculating eigenvectors and eigenvalues of a matrix. Thank you very much!
If learning platforms were as artistic as here, learning would be actually motivating and engaging. I'm highly thankful for this godlike content, though I wish education systems were modern, convenient and not subordinate to perfection.
Thanks for the tutorial. However, I would like to know if the formula applicable to any 3 x 3 matrices?
That's a great question. The short answer is that there's nothing nearly as nice, because for 3x3 matrices you have to solve a cubic equation, and while there does exist a cubic formula, it's not nearly as compact as the (simplified) quadratic formula. However, you can use this, plus an extra step, as a nice shortcut (well, short-ish) to find the characteristic polynomial. If the characteristic polynomial expands to be x^3 + Px^2 + Qx + R, then it's still the case that the sum of the eigenvalues (the trace of the matrix) is -P, and the product of the eigenvalues (the determinant of the matrix) is -R. But now, there's a new invariant of the matrix we need to account for, that linear term Q. If the eigenvalues are L1, L2, and L3, and you think about expanding (x - L1)(x - L2)(x - L3), you can see that Q = L1*L2 + L1*L3 + L2*L3. This is an invariant, just like the trace and determinant, though to my knowledge it does not have a distinct name. The question is, given a matrix, how can you figure out what this is before you know the eigenvalues? Well, if the coefficients look like this: [[a, b, c], [d, e, f], [g, h, i]] Then by taking the time to expand out the characteristic polynomial (subtract lambda off the diagonal and compute the determinant), you'll see that this new invariant is the following: Q = (ae + ai + ei) - (bd + cg + fh) It's a new computation, somewhere halfway between a trace and a determinant. It has a sort of pleasing visual symmetry to it on the grid of numbers, and if you try it for a few matrices you'll see that it's not too bad to write out, a little easier than the determinant. Combining that with the trace and determinant, you can write out the characteristic polynomial of a 3x3 matrix decently quickly. And from there, maybe you're lucky enough that the polynomial can be factored and solved quickly, but otherwise, you're doomed to use the cubic formula. Or, you know, at this scale just pop over to WolframAlpha and just ask for the eigenvalues :)
@@3blue1brown Thanks for the detailed explanation. It helps me understand more about eigenvalues and eigenvectors, which is important for my project. Thanks once again.
@@3blue1brown Thanks so much! Actually just commented the same question, before reading this. Expanding this to 4x4 etc. would probably keep producing more invariants and make stuff more and more difficult I guess.
@@3blue1brown Q is the sum of the 2x2 minors of the matrix. This generalizes to square matrices of any size. (The trace is the sum of the diagonal 1x1 minors; the determinant is the sum of the one 3x3 minor, etc.)
See my general comment above--it involves one more invariant, the surface area. The cubic formula is nasty, though!
Dude, you are just so freaking awesome, you have done what my COLLEGE PROFESSOR couldn't do, and in less time, more efficiently, and with a nice little trick to compute something that was kinda complicated and takes too much time, with a nice jingle with it, really, keep going, you are just the best!
Literally in the last few weeks of my intro Linear Algebra course. This could not have come at a better time!
Another video finally
i passed my linear final bc of u ❤️
Wow the formula at the end was incredible! Was so fun going through the discovery process with that.
we actually learn in quantum mechanics to start computing eigenvalues with pauli matrices and from that moment the traditional way is forgotten. so im glad you posted it, this is very useful
I find it exceptionally pleasing that the matrix made up of the first four digits of e has integer eigenvalues .
Brilliant! Nit: I think it would be easier to remember if you had used "d" for the determinant instead of "p".
D for product?? What madness is this?
now you will have to remember that p is the product of eigenvalues which is the determinant d
Never expected to see a new linear algebra upload after 2016. I'm satisfied.
Glad to see a new video in this series, I have been keenly following your series and repeating it few times to make sure that the content sticks. I would like to take this opportunity, to request of a series on Maths to learn Quantum Computing I have found many videos from this series helpful in understanding quite a few concepts in Quantum Computing and hence the motivation for requesting for a series, as Quantum Computing needs to wrap our minds for the operation in abstract vector space. Thank you once again.
Nice video, but am concern on How to get paid, any help please
Through what kind of job?
Is it online Investment?
@@murraydickson6336 Yes, online investment
I get paid directly to my account
@@murraydickson6336 how please
Me: *still waiting for probability 3*
LOL the jingle got me, it's always interesting how you can solve things two different ways.
This trick is really cool and I wish or maths professor would have the same enthusiasm explaining maths as you do!
what about the eigenvalue of 3x3-matrices? how to compute them?
it only works to some extent. only the calculations for m and p are still correct however you can't actually use the m +- sqrt(m^2 - p) thing e.g. [4 -1 6] [2 1 6] [2 -1 8] eigenvalues = 2, 2, 9 (eigenvalue 2 has algebraic and geometric multiplicity of 2) using the formula, m = 13, p = 36, m + sqrt(m^2 - p) = 9 m - sqrt(m^2 - p) = 4 (= 2 + 2)....eh it kinda works. but in most cases it doesn't work at all e.g. [ 1 3 3] [-3 -5 -3] [ 3 3 1] eigenvalues = 1, -2, -2 (eigenvalue -2 has algebraic and geometric multiplicity of 2 as well) using the formula, m = -1.5, p = 4 but when you do the math.. sqrt(m^2 - p) = sqrt(1.5^2 - 4) = sqrt(-1.75).... whoops.
the computation for 2x2 matrices is actually just computation for roots of a quadratic equation (as seen in another of his vids, search lockdown math episode 1) , because thats the characteristic equation for a 2x2 matrix's eigenvalues. so the computation for 3x3 matrices would be the cubic formula......😪😪
oh i see. would have been easier if there's a shortcut, but i guess it's fine then😄 and thank you for the help!
Literally 3 hours too late, I just had my linear algebra final
😂
Aside from everything explained in an insanely intuitive and reasonable way, i adore the music here :)
Got a differential eqs and linear algebra final next week. Perfect timing, Grant!
I feel personally attacked that you waited until the day after I finished my diff eq class to upload this
Didn't you have L.A. in your first semester? It's the basics of most math that deals with data and dimensions, ortogonalithy etc. It's a 2x2 matrix o.o Do you have diff. eq before Multivariate Calculus? That looks like a weird syllabus.
@@genericnamethingy Yes, I did take it, I learned the "shortcut" that the eigenvalues of a 2x2 matrix, A, can be found from lambda^2 - tr(A)*lambda+detA=0, but this is quicker. And I took diffeq and multi at the same time
@@hudsonmcgaughey6798 You're subtracting a matrix from a value and then adding a value, are you sure about that formula? The standard one is det(A - I*lambda)=0 with I being the identity matrix, this works for n*n
@@genericnamethingy tr(A) isn't a matrix.
ok im gonna be that guy who always says it: *I NEVER CLICKED SO FAST!*
This video came at the perfect time in the semester! I was just doing a problem on my Quantum final exam regarding Super-Symmetry and I needed to compute the eigenvalues of a 2x2 Hamiltonian in a rather nasty sub-space. 3Blue1Brown to the rescue!
Keep it up 3Blue1Brown, this channel explains complex maths stuff as if it is a simple story. Proud of you and of your channel. - Ace In Academy
KZhead: 1 view Also youtube: 66 likes and 11 comments. I don’t think the math works out on that.
refresh
@@monk9008 ik, just a bit funny it does that.
@@mqxf yeah😂
The worst is when they ask you for a 3 by 3….
This literally could not have come at a better time. I've got a calculus exam next week (featuring systems of linear ODEs). Thanks a lot :)
I just finished up Linear Algebra last week, so this won't be of any immediate help. But I will take the chance to say that your series on Linear Algebra made the class much easier and make much more sense, so thank you so much for everything you do.
Thanks for posting this a week after I graduated, perfect timing
This is like learning the power rule for the the first time. It's so quick. Definitely going to use this on my linear algebra final
Been waiting too long for a new 3b1b video. I was on my third cycle through the channel since the last new one.
So glad I stumbled upon this video before my exam tomorrow! Thank you for sharing!
Thank u so much 3b1b. This channel is really cool. I have learnt how to feel math from here.
Eigenvalues and Eigenvectors cenfused soooooo long time. It is the first time I feel that I finally understand them! Thank you so much for the excellent work
I love how this is the exact same as the quadratic shortcut in the lockdown math video 🧎♀️ tysm
Nice! I got flashbacks from lockdown math, it was a memorable moment worth revising.
I am doing a quantum physics course and one about dynamical systems and this video is just awesome. Could not have come more handy thanks!
OMG that jingle is awesome!!!
I like that your videos are really expanding on color. I especially like "eigen-teal" and "variating pink"
Two things : number 1 : I don't think anyone would have just skipped to 4:53 everybody loves to hear you. 2ndly the music Grant's new etude is sooooo beauty. It's piece of peace !
Please post more! I've already finished my math minor, however, I love your insights on numbers!! I don't care if it's stats, linear algebra, calculus, or whatever - more videos!
Thank you, this realization is useful not only for computing eigenvalues but also for theoretical purposes. I recently used it when I tried to describe all 2x2 matrices with some fixed determinant.
Absolutely beautiful callback to the first live lecture last year
Perfect timing! I begin teaching a linear algebra course next week. 😊