4. Eigenvalues and Eigenvectors

MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/18-065S18
KZbin Playlist: • MIT 18.065 Matrix Meth...
Professor Strang begins this lecture talking about eigenvectors and eigenvalues and why they are useful. Then he moves to a discussion of symmetric matrices, in particular, positive definite matrices.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@jrippee05]

Good morning, Dr. Strang. It is always a pleasure to listen to your classes. I wish all classes were as well organized and thorough as yours. It is always a joy to listen to your classes.

[@debraj92]

MIT is MIT for a reason. Thank you for open sourcing such wonderful videos.

[@thiagopbueno]

Special is good. Useful is even better...

[@johnk8174]

22 minutes in, still waiting for the hard part; that's the genius of Gilbert Strang.

[@georgesadler7830]

This is an outstanding lecture on Eigenvalues and Eigenvectors. Eigenvalues and Eigenvectors are very important for solving linear systems especially in differential equations. MIT and DR. Strang thank you so much.

[@atomscott6495]

43:28 Strang sensei thinks student makes a mistake Strang sensei : *Death*

[@yd9939]

せんせい：定番なミスきちゃ！！

[@starriet]

Just one feedback from a student: It would be even better if the camera doesn't move too frequently following the lecturer. Thank you for all the camera works, just wanted to help make them even better. Thanks for great videos.

[@RC-bm9mf]

Thank you very much dear professor Strang. You have been saving and will save so many students.

[@testus86]

I had this in my bachelor of computer science in german. My prof was way worse and he was talking in my language. I understand more this in english than my prof. In my language. Huge compliment to Dr.strang

[@acacianorison]

Great lesson from a humble Professor with a sense of humor.

[@allandogreat]

Love and appreciate Dr. Strang

[@xh3221]

love the professor for clarity. I had no such a teacher in my college education

[@aarifhussain3700]

Blessing to all peoples those are related to mathematics field

[@JulieIsMe824]

Great lecture!!!Thank you Prof. Strang!

[@rob3c]

How lucky we are to have another wonderful Strang lecture! His insightful presentations are always a treat, and it's great to see his take on deep learning applications. Minor chalk-o: he rotated Ax the wrong way at 27:22 (but the math is still right)

[@kellypainter7625]

Even gods make mistakes.

[@yorgunkaptaan]

@@kellypainter7625 No.

[@yizhongsha]

Brilliant, better insight than the original 18.06

[@usmanhassan4498]

Agreed

[@TheDroidMate]

This man is a legend. Thanks for everything.

[@gokulakrishnancandassamy4995]

Thoroughly enjoyed Prof. Strang's lecture as usual (though it pains to see how aging has affected him!)

[@freeeagle6074]

Expecting to see Dr. Strang lecturing at age 106.

[@Enerdzizer]

In difference equation in 11:00 it is better to compare differential equation with v_t+1 - v_t =A* v_t.

[@user-sc2ei2lf9o]

it's a very good course for someone to learn further on Matrixes in bachelor of Computer Science

[@suprithashetty9016]

Way of solving ❤️

[@tusharganguli]

Man! the camera guy has completely messed up such a beautiful lecture!

[@allyourcode]

@22:05 But how do we know B is invertable? I found a proof that does not assume B is invertable: Suppose we have x such that ABx = lambda * x. Left multiply both sides by B: BABx = lambda * Bx. This shows that Bx is an eigen vector of BA, and its eigen value is lambda.

[@suprithashetty9016]

Impressed ❤️

[@snnn_wow]

28:00 Was it rotated to wrong direction? For example, if x = [0,1]^T, then AX = [1, 0]. So it is clockwise 90 degree rotation.

[@learningstatistics1290]

Yes, you are right. A good point.

[@yb801]

Will this course cover jacobian and hessian matrix?Just asking.

[@faroukguituone5296]

good teacher

[@aungkyaw9353]

"vectors from the space formed by independent eigen vectors of original matrix A == eigen vectors themselves for some similar matrices to A (with same eigen values)"? Is this statement true or false? 42:24

[@justpaulo]

I think it's false. Here's why: A = X Λ X¯¹ B = M (X Λ X¯¹) M¯¹ = (M X) Λ (M X)¯¹ so the eigenvectors of B will be M X = [Mx1 Mx2 ... Mxn]. Each column of M X => Mx¡ is a linear combination of the columns of M, therefore it is in the column space of M ( C(M) ), but not necessarily in the column space of X. If the eigenvectors of B turned out to be XM, then they would be for sure in C(X), i.e. they would be a linear combination of the eigenvectors of A.

[@eduardojreis]

I'm very thankful for these lectures. Though, the camera movement is sometimes annoying.

[@seventyfive7597]

Yep, the old camera angles, straight on and more static, were much more reasonable.

[@343clement]

i wish they didnt move the cameras so much, i want to look at the blackboard, i don't mind if the professor is not in frame.

[@igormorgado]

you know that you can pause, right?

[@adaelasm6467]

Yeah and then you aren’t hearing the professor talk about the equation

[@jongxina3595]

At 22:00 M = B only applies if B is invertible right? What about other cases when B isnt?

[@suprithashetty9016]

Solving ❤️

[@anunaysanganal]

Are eigen vectors of a symmetric matrix already unit vectors, or we need to normalize them?

[@zma4543]

we need to normalize them to have length of 1 for each vector to get orthogonal matrix. I found this reference pretty good to answer your question in detail.

[@yujeong8373]

22:00 25:20

[@eduardojreis]

10:56 - Could someone explain this? I didn't get the derivative.

[@matthewearley3518]

Check this link out: math.mit.edu/~jorloff/suppnotes/suppnotes03/la5.pdf He's making a overall comment on how eigenvectors are used to solve systems of linear differential (continuous-time) or difference (discrete-time) equations. It is one of their principal uses.

[@heretoinfinity9300]

Is the equation in 22:00 written with matrices M and M inverse switched?

[@elisad8372]

yes I believe so

[@Fan-vk8tl]

both definitions is the same

[@PrzemyslawSliwinski]

0:45 - We have heard about them eigentimes! ;)

[@suprithashetty9016]

Style ❤️

[@suprithashetty9016]

Duster ❤️

[@suprithashetty9016]

Way ❤️

[@suprithashetty9016]

Golden hair ❤️

[@suprithashetty9016]

Mic ❤️

[@suprithashetty9016]

Board ❤️

[@suprithashetty9016]

Chalk ❤️

[@suprithashetty9016]

Math ❤️

[@suprithashetty9016]

Handwriting ❤️

[@keyboard_toucher]

6:23 "that long, infinite series" hmmm....

[@matthewearley3518]

He is talking about a taylor series of e^(ax) e^ax = 1 + ax + (a^2)(x^2)/2! + (a^3)(x^3)/3! ... + (a^n)(x^n)/n! Since he has already proved that (A^n)*x=(lambda^n)*x, he just has to combine these two properties to prove that e^(Ax)=e^(lambda*x)

[@marcusstoica]

@@matthewearley3518 Thank you--saw the original comment before seeing the video, and came back down to answer it once I knew the context. Only thing I would add is that n -> +infinity.

[@suprithashetty9016]

Jazz ❤️

[@suprithashetty9016]

Accent ❤️

[@suprithashetty9016]

English ❤️

[@moritzstrueve5184]

It`s kind of funny, the word "Eigenvector" is a mix of german with english

[@thangible]

except the german have the word vector too. Eigenvektor.

[@hxqing]

还好。我们不把它译为“爱根向量”，而译为“特征向量”。

[@moritzstrueve5184]

@@hxqing danke dir

[@learningstatistics1290]

25:24 To prove AB and BA share the same eigenvalues, I think here the proof only proves the case when B is invertible. So this is not a general proof.

[@muhammadmubashirullah7152]

oh God the distractions.

[@kevinchen1820]

20220517簽

[@susantabhattacharya6323]

Dr. Strange.

[@rafiaumar7787]

How these eigenvectors and eignvalues are Helpful In Industrial engineering field.....????

[@o.y.930]

u ever heard of google????

[@rafiaumar7787]

@@o.y.930 Yup i know ....Should I prefer GOOGLE to find the answer of this question??????¿¿¿

[@kevintoner6068]

Unwatchable due to random unnecessary camera changes, such a shame. Seemed like it was gonna be an awesome lecture

[@sajalvasal5073]

DEATH ... LOL

[@user-wr4yl7tx3w]

Never seen a worse camera man.

[@reneeliu6676]

I'm the 951 viewer and 2nd commenter!!

[@adaelasm6467]

Please stop taking the camera off the equations!!

[@TankNSSpank]

stop panning the camera! stay on the balckboard