Eigenvalues and Eigenvectors

MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
View the complete course: ocw.mit.edu/RES-18-009F15
Instructor: Gilbert Strang
The eigenvectors remain in the same direction when multiplied by the matrix. Subtracting an eigenvalue from the diagonal leaves a singular matrix: determinant zero. An n by n matrix has n eigenvalues.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@reneeliu6676]

I wish I could someday bump into Dr. Strang in a supermarket because I want to salute to him. A while back when I was learning algebra in college, I was paying tuition to my own professor (didn't learn anything from him) while learning everything from Dr.Strang's older videos for free. I am very grateful until this day and always will be.

[@kiichit1072]

Love those lectures that begin with why you need it

[@DLSMauu]

yeah that is key on why these lectures are so great

[@Zaki_1911]

Same

[@ianbrewer4843]

Same

[@milkdrinker7]

I just learned more in 10 minutes of you than I've learned in the last 2 weeks of lectures

[@naf7540]

Professor I have an infinite admiration for your clarity and precision of your mind. Your lessons are unequalled. Thank you so much.

[@rajvinderkaur1013]

thanks alot sir. i dont know how can i pay gratitude to u.. Thanks to the MIT. great appreciation.

[@zainuddinkhan834]

sir i have never seen a better mathematician than you.

[@jonathansum9084]

You are right.

[@nirbhaythacker6662]

He's good, sure, but there certainly may be more intelligent people.

[@prateethnayak8422]

Intelligence is not a good metric if you cant express it in laymen terms.

[@nkhullar1]

very good professor, no doubt.

[@bonnome2]

He was 80 when he made this video, pretty phenomenal.

[@zhewang4226]

The videos are SO HELPFUL! I had no idea what my professor was talking about during my lecture. Now, I actually understand stuff!!!

[@umutguvercin]

The reason why i love linear algebra is you Sir. Enthusiasm to teach is the most important thing a lecturer should have, and i am glad the say that you have enough for thousands of students. :)

[@MelodySaleh]

So Damn well said about the greatness of his enthusiasm.

[@SparkyElectricDave]

History for generations will remember your good work, just love it.

[@hasanshirazi9535]

Thanks a lot Professor. Your lecture clarified the eigenvalues and eigenvectors very well.

[@Ggnera]

Prof. Strang is excellent at teaching! The video was very useful. Thank you!

[@georgesadler7830]

These are powerful linear algebra concepts. Linear algebra is a power tool in signal and systems theory, which is a part of the electrical engineering program. When I took this class at the University of Maryland College Park years ago there was a little emphasis on linear algebra. Dr. Strang thank you so much for your contribution to the subject.

[@rutika17]

Sir you are awesome. Can't find a better teacher than u. Thanks a lot for all your efforts.

[@mustafasoylemezo5694]

İ think his explanation is so clear and fluent.İ like his lectures very much and i appreciate him.

[@erion3017]

That was a gold lecture , in 19:00 minutes i learn what my teacher was trying to tell 4 lessons!!!

[@joefagan9335]

My word!!!! Every sentence is precious 💝💝

[@FloppyDobbys]

It makes intuitive sense that the Eigen vector remains the same with A^n because we can see A^n as just applying the same transformation n times. Applying the same n times doesn't change the direction of the eigen vectors. For example, If we apply a sheer matrix 3 times to a vector. This doesn't change the eigen vector direction but it will change the magnitude of the sheering because we apply it three times rather than one time so of course the eigen value associated with the composition of all three together must be the same magnitude change if we applied sheer as three separate transformations.

[@fisicaematematicacomjean]

I have never tought about it. Thank you very much for blowing my mind up haha. Math is all about intuition, I think. However, sometimes it's hard to really see through the mathematical expression. Thank you, Tyler.

[@isaaclara5547]

Amazing lecture. Thanks MIT OCW!

[@davidwright8432]

Many thanks, Gil! If all math instruction were as clearly and carefully explained as yours, math would be a lot more popular - because people would realize it was something they could do - like riding a bicycle!

[@AJ-et3vf]

Awesome video lecture sir! Very insightful and enlightening!

[@684tranminhtuan]

Thank you very much for your kindness to provide a lively and wonderful instruction.

[@emanmagdi216]

I actually enjoy your videos so much thanks a lot sir. I wish one day to attend one of your lectures

[@ashishneupane1457]

That's why everybody wants to get into MIT. My professor needs to see your lectures.

[@salrite]

Another Great Explanation by Prof.!!!

[@ivanordonez1183]

Excellent explanation!! Thank you.

[@sergiocontreras4k]

It is impressive how to this day this knowledge has not been lost, I mean, KZbin videos are always difficult to watch after years, not this one, it is just as good today, that when it was done

[@rinrin6454]

Hahaha "That's the big equation, it got a box around it."

[@gangaprasadyadav2194]

amazing concept and explanation

[@lapertica8258]

THANK YOU SIR GILBERT

[@richardgraziano428]

Gilbert Strang is absolutely brilliant!

[@yunzheli8784]

I love this man so much!

[@realityandphilosophy4912]

This professor is amazing!

[@kunleolutomilayo4018]

At 2:55, there I lightened up! Didn't see that coming from previous derivations.

[@goPistons06]

the introduction is quite elegant and informative, but so simple at the same. such mathematical beauty.

[@goPistons06]

plus such great pedagogical skills. It makes it all come alive. Congrats to the teacher. Cheers from Chile

[@abdallahallahham8586]

Dr Strang you are really the best

[@ANunes06]

"okay." is a better youtuber intro slogan than 90% of the market.

[@umeng2002]

Thanks. My Finite Element Analysis professor blasted through eigenvalues and eigenvectors a bit too quickly.

[@TheDavidlloydjones]

This is a brilliant video: he wrote all the stuff on the board before he started the camera. The guy's a freak! This will horrible shock to all the people who think that KZbin is a technology for showing the back of people's heads as they scribble on the blackboard, but so be it. The shock will loosen them up for the other one to come: the guy doesn't talk for ten minutes and then say "without further ado, let's get started." This Strang guy has shown us the etymology of the word "strange."

[@ernstmasseant8659]

Great explanation

[@shivanshuraj7175]

You are so good mathmatician.!

[@martinepstein9826]

Oooh so any polynomial expression of A will also have the same eigenvectors, and the'll be of the form P(A)x = P(lambda)x. Nice!

[@MrCigarro50]

Best wishes to such super-profesor.

[@joebrinson5040]

When you really want to know then you watch Dr. Strang!

[@piyushverma2074]

You are great sir..

[@ritual17]

Thanks Gilbert, my little friend

[@annakh9543]

this man is unbelievable

[@learninfact9281]

Surreal to think like that...

[@andrewl5267]

Do you get the same eigenvector multiple times when an eigenvalue has an algebraic multiplicity greater than one?

[@shivanshuraj7175]

Thank u professor!

[@matinhewing1]

Gilbert...a God amongst men...

[@c0t556]

Aesthetic Athlete YES

[@ihbarddx]

I love this guy! He's somewhere between a math professor and Mr. Rogers.

[@TomSkinner]

haha, good characterization . I love this guy.

[@raminarezou1009]

Easy explanation,

[@fernandoraphael95]

He's the best

[@kwccoin3115]

Wow. Think I got it. So good s lecture.

[@raphaellungu2424]

l love your teaching

[@simonelgarrad]

can I know when is this the next video coming up ?

[@mitocw]

+simonel garrad Here is the playlist for this series: kzbin.info/aero/PLMsYJgjgZE8iBpOBZEsS8PuwNBkwMcjix and here is a link to the course website: ocw.mit.edu/RES-18-009F15.

[@simonelgarrad]

MIT OpenCourseWare thankyou...so much :')

[@MS-dr9et]

Ax=lambda x Dats the big equation. It got a box around it ! :)

[@qtmomo]

Can someone explain to me how he did work in advance? If you follow 18.06 the chain of thoughts is kinda opposite or did I get something wrong? You first find the eigenvalue then you plug in to find the eigenvector right?

[@carultch]

You find the Eigenvalue first. Then you plug it into the diagonals along the given square matrix, and multiply that square matrix with the eigenvector as a vertical matrix. Equate it to the zero vector as a vertical matrix. This will create a system of equations with at least one of them being redundant. Let one of your terms of the Eigenvalue be 1 or any other convenient number, and solve for the remaining terms. Then you'll have your eigenvector corresponding to that eigenvalue. Repeat for the other eigenvalue(s).

[@tasmiahtunazzina6184]

may I know how the solution of the differential equation was y= Ae^*t x?

[@carultch]

It's what's called the Ansatz solution, or as I like to call, the prototype solution. It's a solution form we assume, because of experience with the exponential function and its favorable features when it comes to differentiation.

[@nguyenbaodung1603]

God please make this man immortal

[@bhaumik09]

#Respect

[@alexandraboehmke2102]

How does the n at 15:30 relate to time dependence? I don't see how each time step is another equation.

[@simonelgarrad]

Alexandra Boehmke yeah what does it mean that all the time dependence is in the exponential??..

[@Lolwutdesu9000]

simonel garrad because the t is in the exponential. X is a variable that doesn't depend on time.

[@materiasacra]

Look at the original differential equation: dy/dt=Ay. Think of dt as a finite tiny time step, and rewrite: dy = Ay dt. This says that the change in y over the course of time step t->t+dt is proportional to the duration of the time step and Ay. So the matrix A 'generates' the temporal change by operating on y, for tiny time steps. In physics we call i times A the 'Hamiltonian' of the system. (The i is there for convenience in a wider context.) Now what about the evolution over a longer time interval [0,t]? We split it up into n tiny steps of duration dt = t/n, and apply dy = Ay dt over and over again: y(0+dt) = y(0) + dy = y(t) + Ay(0) dt = (1+Adt) y(0) y(0+2dt) = y(0+dt) + Ay(0+dt)dt = (1+Adt)^2 y(0) y(0+3dt) = y(0+2dt) + Ay(0+2dt)dt = (1+Adt)^3 y(0) .... y(0+ndt) = y(0+(n-1)dt) + Ay(0+(n-1)dt)dt = (1+Adt)^n y(0) That last equation can be written y(t) = (1+At/n)^n y(0). Here you have the n-th power of a (scaled and shifted) matrix A determining the time evolution of y. The formulation in discrete time steps may or may not be the most convenient. If we want, we can take the limit of n -> infinity, thus making the steps dt arbitrarily small while having arbitrarily many of them in inverse proportion. Then we find: y(t) = e^(At) y(0). This is all well and pleasing to the eye, but if you ask: what does an exponential function of a matrix MEAN, we have to revert back to the series expansion of the exponential: y(t) = sum_j (1/j!)(At)^j y(0) which takes us right back to powers of a matrix.

[@rutika17]

👍👍

[@simsim2159]

Teachers suck here It was good listening to you

[@Richinkind123]

He is genius 😢

[@idealpotatoes]

This sir is the jesus of algebra

[@xiangyanggao2946]

Professor Gilbert said, suppose we just found these two eigenvectors with your naked eyes, then...... well, interesting.

[@kattasudhir]

I wish Prof. Strang would give a intuitive meaning of eigen vector and eigen values before dive deep into math of eigen value and vectors

[@omkarchavan2259]

watch this video

[@kattasudhir]

Where is the video

[@mayurkulkarni755]

watch MIT 18.06

[@ernstmasseant8659]

He says that at the middle of the lecture by saying it is useful for thing that moving over the time .

[@joebrinson5040]

That is this video!

[@haydenullrich2388]

(Gilbert)X = (strang)X

[@thomasfranzstockhammer7846]

Lg

[@rafikzorrik752]

اساعة تراقب العالم

[@konradmtb]

Kind of messy and lacks the insight which is critical. Not even close to what is shown by 3blue1brown....

[@vp4744]

The difference is due to the preparation of the audience: one is made for MIT students and the other is made for community college students.

[@ShauriePvs]

Insight videos like 3blue1brown are impossible to be taught in black board in class.

[@salrite]

Another Great Explanation by Prof.!!!