21. Eigenvalues and Eigenvectors
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- Опубліковано 8 тра 2024
- MIT 18.06 Linear Algebra, Spring 2005
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/18-06S05
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21. Eigenvalues and Eigenvectors
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Audio channels fixed!
Thank You.
Thank you.
Thank you MIT
Thank you!
Thank You
A lot of viewers here stating that they wish they had a lecturer like Gilbert. I am one of those viewers. But as a lecturer myself, I sometimes find myself in a room full of students who don't give a sh.t about anything I say or anything I explain, and it comes to a point where I want to quit my job. When you click on a link to learn math you are fully motivated. Gilbert has a full room of MIT minds who are open to learning. But when you face a group of students who are just looking for loopholes in your syllabus, who don't give a sh.t about your lessons, always fight for 0.01 points every time you conduct an exam, you began to question things. Yeah, I wish I had a lecturer like him and I am equally sorry that I cannot be a lecturer like him.
You should need to be sorry for it. You are a lecturer, if you like to do so, keep on improving. Somewhere, somehow you will come across some of students who are opening to learning. That is enough, one student is enough
There is a good chance that someone will be interested in the lesson. It might just be difficult to spot these students especially when students are scared of asking or answering questions in lecture halls. I feel so bad whenever lecturers ask a question but no one answers. It makes it seem like everyone is not interested, but when in reality at least one person will want to learn. I guess if I am a lecturer myself, that is reason enough to continue teaching. :).
@@mori1799
Honestly, I believe that one major reason why a lot of students hesitate to ask questions during lectures is because a lot of people pull the "stupid question" move as soon as someone asks a question that happens to have an answer that is familiar for most people;
this makes students feel like "maybe I shouldn't ask this question, because I will seem stupid to everyone else if they happen to know the answer to that question".
We should encourage people to ask questions whenever they are looking for the answer to that question, _even_ if it might appear to have an "obvious" answer to most people.
Yes, of course there exist rude and highly inappropriate questions that aren't okay to ask, and sometimes you are required to know the answer to certain things (for example, a pilot had better be able to answer questions about the basics of how to control an aeroplane), but if someone asks an honest, harmless question, then it also deserves a polite answer.
@Peter_1986 I agree. Students tend to overestimate the knowledge of their peers. I had a lecturer once who told us that if you have a question for something, then likely the majority of students in the hall will also contemplate the same question. In my experience, I remember always hesitating to ask a question, because I would just assume that I forgot a specific Lemma or a theorem which has been used in the proof. I didnt want to interrupt the lectures too much, so I'd rather work through it on my own.
Math is a rough subject to lecture for. Unless you’re out of the gen Ed area, you’re guaranteed low motivation from students. Math is one of the least appreciated school subjects in the west.
1. What are eigenvectors and eigenvalues? @00:00
2. Example 1: Eigenvectors of Projection Matrices @5:24
3. Example 2: Eigenvectors of Permutation Matrices @10:42
4. How to find eigenvalues and eigenvectors @16:35
a. Det(A - lambda * I) = 0 @16:45
b. Finding eigenvalues @23:25
c. Finding eigenvectors @26:20
5. A + 3*I has same eigenvectors as A while the eigenvalues added by 3 @29:37
6. Eigenvalues(A + B) != Eigenvalues(A) + Eigenvalues(B) @32:53
7. Eigenvalues of rotation matrices @37:00
a. Complex eigenvalues @42:04
8. Eigenvectors of shear matrices @45:42
!=
A fellow programmer I see
Great bro
Good job
Lol I know how to do computations with eigenstuff and thought I didn't need this lecture. Then within the first 5 minutes or so, he starts discussing eigenvectors in terms of projection matrices (if you're already in a subspace of the projection matrix, your direction doesn't change). This kind of esoteric insight is why I've followed this lecture series from lecture 1. He gets my head turning every time.
Professor Strang, you are a gift to humanity, I wish you well! And many thanks to MIT OCW for making these teachings available to people everywhere ☺️ A lot of good will come of this
Anyways, back to the lecture.
OMG same. I wish my professor was like Gilbert
omg, yeah! I did the exact same path as you!
Me tooooo
where are other lectures
As a German I'm obliged to note that: "Eigen" is a German word and means "own" or "self".
(edit was 'cause of a typo)
Thank you so much that makes sense
My textbooks translate eigen as "characteristic".
@@starguy2718 Yeah. This might be more accurate overall, but it doesn't give you the same feeling that germans have when hearing "Eigenvektor". We have the word characteristic aswell in German ("Charakteristisch") and you can say "Du bist mein Eigen" meaning "you are my own", not "You are my characteristic". So it depends. Im no language expert but Eigen and own might even be descandents from the same word, because they some(edit: sound not some) similiar and basicly have the same meaning.
@@luojihencha That was what interested me some years ago. I knew it was a German word. It made it seem more fascinating. I'm not very good at maths. I knew from some telecom etc and electronics that this maths was used somehow to decode noise in signals (or find information in a distorted signal). But I had no idea how. I didn't need to know. But it intrigued me. I am a writer and poet not actually in any science field but keep some interest. But I find this maths as he does it like something more than maths, it is like a strange abstract poem and yet it is illustrated by referring to vectors and vector spaces which remind me of Wittgenstein's 'logic spaces'. W started as a mathematician but became a philosopher.
It is almost as if the whole thing for me was about the word 'Eigen' as I sounded it. (Probably not as interesting to a German speaker! Just the way I imagine it...)
Gilbert Strang is a gift to humanity!! This dude single-handedly made me love one of my most hated classes from college.
Cheers from Brazil 🇧🇷🇧🇷
Hi from Brazil
He is truly gifted.
Listening to him think his way through problems is the analytical education we need, and not just for algebra. His candor is well appreciated. He attacks very complicated ideas with a simple tongue. And in spite of all the newer technology, i am learning more from his approach than the newer educators.
I agree he is a great teacher, but honestly these are not complicated ideas, this is basic linear algebra
'if i had a prof like him' statement in my head
If anything stands out from these videos is the teachers humbleness - if this word exists.
They are modest and clear and respectful. And this is so rare these days.
Thank you, MIT!
They are simply bored. They have done this stuff a hundred times before. :-)
Humility is the word you're looking for.
Best lecture that explains intuition behind eigen values and eigen vectors. It's not just about cramming the formula.
I can gurantee that if you watch it once, you will never forget. Literally, explanation is at awesome level with easy language.... Hats off Prof. Gilbert Strang❤️
From this great lecture, I am finally learning the full meaning and understanding of eigenvalues and eigenvectors in linear and system theory. DR. Strang is the leading commentator on this subject.
I've looked at a lot of explanations for Eiganvectors and eiganvalues but this is the first one to clearly identify the vectors in R3. and to show how zero lambdas represent vectors in the null space.
This was a topic that I found more difficult at University. Had to pause a few times to follow and understand correctly but it was worth it. Really good explanations here.
At 1:05 he says "What does a matrix do? It acts on vectors". I just love the way he looks at the matrices. He is definitely a "matrix" artist. Every time I watch his video, I get a feeling that I learnt something new. Absolutely brilliant sir!
Amazing and intuitive material, keep these coming.
Dear Gilbert Strang!! I wish to kiss your hands, your sacred hands as you are my Guru of Matrix Algebra. You explain so well. your way of teaching is miraculous.
Thank you for existing Gilbert Strang. I am grateful to you for showing me the beauty and elegance of mathematics
He has clearly deep understanding about what is he saying...excellent lecture!!🇧🇩🇧🇩
hi vai
Gilbert Strang is an absolute gem.
The way Prof. Strang explains how det(A-λI) = 0 comes is so enlightening! Wish I had taken to this course earlier.
The depth to which Professor Gilbert delves to explain any topic is truly impressive, making the learning experience engaging, comprehensive, and incredibly enriching🔥
Hats Off Professor, You are an inspiration!
Sir Gilbert Stang is my inspiration for learning linear algebra
Professor is gift to the mankind. Thank you so much.
1:39
Maths is not limited to formulas and algebra, it's about insights and thought process, and that makes it beautiful. Thanks Prof Gilbert Strang for wonderful lectures
I love this guy
professor, you are great. I really enjoyed your lectures and will enjoy and reference them later on. thanks a lot
So much comprehensive lecture I've ever seen!!!
Did a crash course on Lin Algebra that purely calculated these. I knew all the steps. But did I really know it? Now i do.. Thank you Gilbert Strang. I endeavour to do ALL your courses that you have available on the internet. What a remarkable world we live in today..
listening to professor Strang is like watching a sci-fi movie, time to time i start seeing the exact same concept in different view points.
I am so awed at the depth of maths and the presenter
Man .. he is the final authority on linear algebra ...
Thank you Camera man too !
This lecture invoked me much inspiration!!
교수님 연금 달달하십니까
Can't help thinking: If Jimmy Stewart taught linear algebra. Excellent lectures.
God bless you, Professor Strang.
"What does a matrix do?" That is the most influential question that I've ever heard in mathematics, and with an easy to follow answer as well! Dr. Strang is definitely the gold standard at teaching this beautiful subject.
Notes:
32:00 If a matrix A is added by a multiple of Identity, and its eigenvalues will turned out to be the multiple number + A’s eigenvalues.
(A+B)x != (a+b)x because we cannot sure if x is the eigenvectors for both A and B, works the same for (AB)x
39:00 With an example of orthogonal rotation matrix:
1. Trace has to be the sum of the eigenvalues and the det has to be the product of the eigenvalues.
2. There is a trouble in orthogonal rotation matrix since we don’t have any eigenvector parallel to the Ax. The eigenvalues will appear to be the imaginary number.
46:20 With an example of a a11 = 3, a12= 1, a21 = 0, a22 =3 (Triangular matrix)
1. Eigenvalues will appear to be in the diagonal.
2. The number of eigenvectors could be less than the the number of eigenvalues
4:20 If A is singular then $\lambda=0$ is an eigenvalue
15:00 An nxn matrix will have n eigenvalues and the tr(A) will be the sum of the eigenvalues
19:30 “A repeated lambda is the source of all trouble in 18.06”
25:00 Shows the polynomial contains the trace and the determinant, in the 2x2 case.
37:15 Example: Rotate vector by 90 degrees
38:47 The determinant is the product of the eigenvalues.
44:20 If we stick to symmetric matrices, or close to symmetric, then the eigenvalues will stay real.
46:50 With a triangular matrix, we can read the eigenvalues off of the diagonal
Thank you Professor! You are truly god's gift to all of us! What a great worthy life!
My favorite linear algebra teacher!
Excellent lecture. I couldn’t miss word he said. Btw, I am a stem phd and took linear algebra class 39 years ago.
This discussion has an application for the quantum sciences. The quantum Hall spin-polarized effect has subatomic particles translations through crystalline structures in arbitrary space. This has an epic important for finding new medical, electronics, energy, and other devices that will change our world.
Years ago I decided to study Linear Algebra. I done a Cert in Engineering and we hadn't done many matrices or enough. So I was really struggling. But I passed as I used the formulae (the course book I had was for Stage 2 and I should really have being Stage 1. But for all that time, out of curiosity I wanted to know what they actually were. I couldn't easily see then how they were vectors. The lecturer makes it clear. I could follow up to finding the eigenvalues. I had always been puzzled how they ignored the other numbers not on the diagonal. Trying an identity as he had and changing the numbers I realised what that diagonal - the trace he called it - was. Amazing! I realized that the 3 1 / 1 3 was 3 x 1 1 / 1 1 It doesn't change except in the amount, not the direction. But in effect only the diagonal from top left to bottom right are significant.
this man is unreal happy retirement legend
This professor is a pure legend
I never seen an mathematical lecture this much interesting.
Jackpot unearthed....blessed
Great teacher. Thank you very much
A is singular -> eigenvectors are in null space.
31:43A에 I를 더해도 eigen vector은 그대로
In two minutes and fifteen seconds, Prof Strang taught me more about Eignenvectors than my lecturer could manage in two hours.
thanks for everything, strang
I hope that beyond the year 2050 somebody finds a way to upscale this masterpiece of lecture series to 1000K 3D holograms.
of all the things on earth, why tho?
Love for Prof Strang .😊. Real picture of Eigen value and eigen Vector ..AX parallel to X ...touched and fascinated with real story behind Mathematics. Lucky students of MIT who is quietly listening the story interestingly.
I really got shocked when I listen to this fact, I wanted to cry !
also, the fact that the Trace(A) = sum of EigenValues, I was like, why did not I have someone to say this before 5 years ?
@@ath216 Late is better than never .✌😊
MIT has scrupulously removed the audio of the audience.
If you are not merely interested in eigenvalues and eigenvectors, you would have noticed the fact in twenty videos.
Try not to make yourself ignorant. This is a good way to add your eigenvalue.
prof.strang teaches you how to think !!! a rare thing in many
A brilliant lecturer.
I was lucky enough to do this at school before going to university. Brings back memories.
God, I love the way you teach!
this is awesome.
Rotation vector Eigen 0,1 -1,0 as Q swing between cos theta and sin theta .
Amazing lecture. Thanks from Italy
Eigen vector direction may change between the same and opposite directions but the eigen value may be increased as a conclusion message of professor.
Very good explication !
Thanks from Italy 😊
always wholesome when Dr Lang interrupts his lectures with a "may I?" or a "shall I?" 😊
Strang
Wish this guy taught me Math 293 and 294 at Cornell. My guy could barely speak English, let alone explain what we were trying to accomplish. I understood that if we wanted eigenvectors perpendicular to x we'd get lift relative to flow...but this guy would have made the math a bit simpler.
Just watched christopher nolan movie... Super lecture..
Like that he starts with eigenvectors and their intuition
The only question I'm now left with is what the hell is "a real New England weekend".
a weekend in new england in america, where MIT is
@@brandonnoll5527 Wait, that implies that there is also a Complex New England weekend!!!
@@adamturian6114 That implies the existence of an imaginary New England weekend too. :)
21:11 there is a subtitle error, he doesn't say anything about complex numbers. It could cause confusion for people who are deaf.
A thousand years from now, people will still be learning matrix algebra from this guy.
explanation is amazing
Best explanation ever I have seen
How admirably clear.
yep
01:39 specially interested vectors - eigenvectors
The next lecture can be....pure happiness. :D
I love Gilbert
just good for the future
its been almost a decade, but I remember he wrote the textbook we used.
Thank you so much! This was so useful and easy to understand!
Thank you!🙂
Basic concepts of streamlining automation. Conveyors systems.
10:41 you're a rock star
thank you sir
Symmetry is dual to anti-symmetry.
Bosons (symmetric wave functions) are dual to Fermions (anti-symmetric wave functions) -- atomic duality, the spin statistics theorem.
Bosons (waves) are dual to Fermions (particles) -- quantum duality.
"Always two there are" -- Yoda.
Good mathematics teachers are a gift from the gods.
Zero X -- Zero interest 18:18
that cracked me up ahaha
Thank you 🦋
수강 날짜: 21 Oct
If you invert a tangential eigen value relative to the elliptical transfraction of its angular differential momentum are you going to end up with an algebraic postulate equal to the original eigen value or will it invert the transfraction into a post eigen mutation?
This has always bothered me.
The math, as I'm sure you'll agree, is not elegant - but is, at least, functional.
Eigen (e)
e~n× t4.046 = a2 + a2r + a3 + a4 ÷ (integer extrapolate) ie 44 × Aa + aA2 (t4 f44)
thank you!
Eigen in Thamizh I Head kann eye how eyes fit into head properly ie eigen vector and eigen value.
Absolutely well done and definitely keep it up!!! 👍👍👍👍👍
😐
this is the big stuff
Oh, I didnt know Peter Crouch also interested in Linear Algebra!
So nice and educative sir
What's the sum of the two eigen values? Just tell me what I just said. :D
Trace
35:24 the moment of truth
I went to a community college and my instructors were at least as good as any of the instructors I've seen on this channel. Specifically my physics instructor, Dr. Lecuyer, my math instructors Dr. Madson and Mr. Young.
useful in engineering on Tall Building!
surprised no one said that (3 - lambda)^2 -1 = 0 => lambda1 = (+1 + 3) = 4, and lambda2 = (-1 + 3) = 2, no need to factorise anything.