A quick trick for computing eigenvalues | Chapter 15, Essence of linear algebra 

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.

😂😂😂

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

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 😁

Says a lot about the mechanics of learning!

Especially when we see it at the same time

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!!!

He began the suffering of every physics students

Thanks, I like learning things

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.

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.

*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

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 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)

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

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 😂

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".

hi

Username checks out

Does your foot get even bigger sometimes ? if yes by what eigenvalue ?

Then you should use matlab bro

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.

Yup, in Vietnamese education system, its called "Vi-et" theorem, probably borrowed from USSR guys

Don't do a PhD just because of some RU-vid 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.

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!

Well when I studied linear algebra came up in a bunch of later courses, even eigenvalues.

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!

Through what kind of job?

Is it online Investment?

@@murraydickson6336 Yes, online investment

I get paid directly to my account

@@murraydickson6336 how please

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

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!

😂

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.

refresh

@@monk9008 ik, just a bit funny it does that.

@@mqxf yeah😂