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
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.
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.
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!
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❤️
@@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...)
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.
The depth to which Professor Gilbert delves to explain any topic is truly impressive, making the learning experience engaging, comprehensive, and incredibly enriching🔥
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!
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
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.
"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.
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..
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.
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
Thanks to the internet, people like us-even from the remotest corners of the world, where access to such world-class lectures was unimaginable during our university days-can now virtually attend these classes and savor every insightful point made by the professor. Thank you!😊
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.
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
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 ?
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.
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.
I'm unclear about the following statement 04:21 "[Singular] means [matrix A] takes some vector X into zero". I know if matrix A is singular, it means matrix A cannot be inverted. However, I don't understand how multiplying vector X by a singular matrix A will result in zero. I was wondering if anyone can help me understand this.
I think Prof. G.S didn't mean that A takes "all" the vector x, but "some", into zero. In my opinion, he just wanted to emphasize that in case the matrix A is singular, we know that the null space is not only the zero vector. Hence, with that knowledge, we can find a special eigenvalue = 0.
If a matrix satisfies the equation Ax=0, that means some linear combination of the column vectors gives the zero vector. In the equation Ax=0, every entry of the vector x is telling you how many of each column of A you need to take (or by what scalar to multiply them), to then add them together and get the 0 vector. If A is a 2x2 matrix, and x is a vector with entries x1 and x2, then you need x1 times the first column of A + x2 times the second column of A. What this is really is asking, is what linear combination of columns 1 and 2 will give the 0 vector. And if there exists a vector x that will satisfy this equation, that also neccesarily means that those column vectors must be linearly dependent - that's how linear dependence is defined. Now if those column vectors of a matrix are linearly dependent, then it also means det A = 0, which makes the matrix singular.
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.
but i am still confused as to where and when at all would we encounter such a situation where we would be required to find eigen values and vectors ?? any domain aor example in mathematics , any example , where we would encounter situations like AX=LAMBDAX ?? Why at all we had to develop a whole subject of eigen , what was the trigger ??
If you are buiilding an efficient ANN, PCA would help. For this an understanding of eigenvalues and eigenvectors of the correlation matrix would help. It would help you to go behind the numbers.
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.
42:40 Hahah are complex numbers that bad? As a passerby (who still watched the whole thing) imaginary numbers don’t seem to bad, neither do eigenvalues/vectors but i’ll admit i can’t answer any example questions yet so maybe i’m just too confident for my own good lol
you are right, but in this lecture we are just looking for real vectors x such that Ax = lamba*x. We used complex numbers for eigenvalues not eigen vectors
Yes, the diagonals of U would give eigenvalues. However, the eigenvalues of U would be different from the eigen values of A because elementary row operations, E, change A. You would have to do EAx = lambad*Ex
We are interested in non trivial solutions, x to (A - lambda*I)*x = 0. For nxn matrix, if A is non-singular, the null space only contains the zero vectors (we are not interested in the zero space). Therefore, (A - lambda*I) needs to be singular to have non zero vectors, x in the null space
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)
When you form the characteristic equation Q-lambda * I You get lambda**2 + 1 = 0 Which has complex roots, hence for a real vector space, we can’t find the eigenvalues of this matrix
why at the last example there is no second eigen vectors .. why it can't be equal {-1 , 0 } .. is that because it will be // to it in the other direction .. and there is no one perpendicular to it ??
I think you might be thinking of a Euclidean space. A vector space does not have an origin. It's the set of tangent vectors (translations) of the elements (points) of a Euclidean space (not a mathematician, just trying to remember the basics here). It is much better to learn to think about vectors as translation (and in some cases as rotations) than to pretend that we are in a space with an origin. That's the actual physical meaning of vectors.
all vectors spaces needs to have the zero vector. This is because to be a vector space, it must be closed under addition and multiplication of scalars. So we have to be able to multiply by zero and still be in the vector space
Audio channels fixed!
Thank You.
Thank you.
Thank you MIT
Thank you!
Thank You
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
400th like kr diya mere desi bhai ❤😂
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
you are absolutely right, a good teacher tells his feeling (or his approach to) about the thing rather than just the thing.
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.
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
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!
Best lecture that explains intuition behind eigen values and eigen vectors. It's not just about cramming the formula.
'if i had a prof like him' statement in my head
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❤️
He has clearly deep understanding about what is he saying...excellent lecture!!🇧🇩🇧🇩
hi vai
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...)
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.
The depth to which Professor Gilbert delves to explain any topic is truly impressive, making the learning experience engaging, comprehensive, and incredibly enriching🔥
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.
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
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.
1:39
"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.
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..
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.
In two minutes and fifteen seconds, Prof Strang taught me more about Eignenvectors than my lecturer could manage in two hours.
I am so awed at the depth of maths and the presenter
Man .. he is the final authority on linear algebra ...
What a great and respectful gentleman, I fell in love with how he see the matrices and those stuff. The world needs such guys more and more.….❤❤❤
Gilbert Strang is an absolute gem.
Thank you for existing Gilbert Strang. I am grateful to you for showing me the beauty and elegance of mathematics
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
Thanks to the internet, people like us-even from the remotest corners of the world, where access to such world-class lectures was unimaginable during our university days-can now virtually attend these classes and savor every insightful point made by the professor. Thank you!😊
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.
Thank you Camera man too !
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.
Teaching cannot get better than this! Can't thank you enough Prof. Strang and MIT OCW! Wish you the best!
Sir Gilbert Stang is my inspiration for learning linear algebra
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
The way Prof. Strang explains how det(A-λI) = 0 comes is so enlightening! Wish I had taken to this course earlier.
I know right! Feel the same way.
Can't help thinking: If Jimmy Stewart taught linear algebra. Excellent lectures.
always wholesome when Dr Lang interrupts his lectures with a "may I?" or a "shall I?" 😊
Strang
21:11 there is a subtitle error, he doesn't say anything about complex numbers. It could cause confusion for people who are deaf.
Hats Off Professor, You are an inspiration!
Amazing and intuitive material, keep these coming.
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.
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?
Excellent lecture. I couldn’t miss word he said. Btw, I am a stem phd and took linear algebra class 39 years ago.
Professor is gift to the mankind. Thank you so much.
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.
I love this guy
professor, you are great. I really enjoyed your lectures and will enjoy and reference them later on. thanks a lot
prof.strang teaches you how to think !!! a rare thing in many
I'm unclear about the following statement 04:21 "[Singular] means [matrix A] takes some vector X into zero". I know if matrix A is singular, it means matrix A cannot be inverted. However, I don't understand how multiplying vector X by a singular matrix A will result in zero. I was wondering if anyone can help me understand this.
I think Prof. G.S didn't mean that A takes "all" the vector x, but "some", into zero. In my opinion, he just wanted to emphasize that in case the matrix A is singular, we know that the null space is not only the zero vector. Hence, with that knowledge, we can find a special eigenvalue = 0.
If a matrix satisfies the equation Ax=0, that means some linear combination of the column vectors gives the zero vector. In the equation Ax=0, every entry of the vector x is telling you how many of each column of A you need to take (or by what scalar to multiply them), to then add them together and get the 0 vector. If A is a 2x2 matrix, and x is a vector with entries x1 and x2, then you need x1 times the first column of A + x2 times the second column of A. What this is really is asking, is what linear combination of columns 1 and 2 will give the 0 vector. And if there exists a vector x that will satisfy this equation, that also neccesarily means that those column vectors must be linearly dependent - that's how linear dependence is defined. Now if those column vectors of a matrix are linearly dependent, then it also means det A = 0, which makes the matrix singular.
A thousand years from now, people will still be learning matrix algebra from this guy.
this man is unreal happy retirement legend
I never seen an mathematical lecture this much interesting.
My college professor just wrote the Av = lambdav and did proofs for 4 hours. Never a single example of what is an eigenvector :(
Thank you Professor! You are truly god's gift to all of us! What a great worthy life!
A is singular -> eigenvectors are in null space.
31:43A에 I를 더해도 eigen vector은 그대로
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. :)
What's the sum of the two eigen values? Just tell me what I just said. :D
Trace
This professor is a pure legend
So much comprehensive lecture I've ever seen!!!
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.
Great teacher. Thank you very much
01:39 specially interested vectors - eigenvectors
Like that he starts with eigenvectors and their intuition
but i am still confused as to where and when at all would we encounter such a situation where we would be required to find eigen values and vectors ?? any domain aor example in mathematics , any example , where we would encounter situations like AX=LAMBDAX ?? Why at all we had to develop a whole subject of eigen , what was the trigger ??
If you are buiilding an efficient ANN, PCA would help. For this an understanding of eigenvalues and eigenvectors of the correlation matrix would help. It would help you to go behind the numbers.
Guys I didn't get the last example. Isn't there an infinite number of eigenvectors? [1,0], [2,0], [3,0], [inf,0] etc?
yes, but we are interested in the basis which is one dimensional. the space spanned by the basis is c[1,0]
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.
My favorite linear algebra teacher!
42:40 Hahah are complex numbers that bad? As a passerby (who still watched the whole thing) imaginary numbers don’t seem to bad, neither do eigenvalues/vectors but i’ll admit i can’t answer any example questions yet so maybe i’m just too confident for my own good lol
check out complex analysis
Can anyone tell me why the professor says A repeated lambda is the source of all troubles in 1806 ? What happened?
I was lucky enough to do this at school before going to university. Brings back memories.
Eigen vector direction may change between the same and opposite directions but the eigen value may be increased as a conclusion message of professor.
10:41 you're a rock star
God, I love the way you teach!
A brilliant lecturer.
Prof. Gilbert is more energetic than me :)))
Rotation vector Eigen 0,1 -1,0 as Q swing between cos theta and sin theta .
Just love to watch your lectures ❤
This lecture invoked me much inspiration!!
교수님 연금 달달하십니까
I have a question, shouldn't the [ i, 0]T, also be a eigenvector for the last example? since Ax=3x for x=[i, 0]T case?
you are right, but in this lecture we are just looking for real vectors x such that Ax = lamba*x. We used complex numbers for eigenvalues not eigen vectors
Who the fuck still uses a blackboard in this day and age? Someone as absolutely legendary as this guy. Old school is best school.
its been almost a decade, but I remember he wrote the textbook we used.
If we make a matrix A to U would diagonal numbers give eigenvalues?
Yes, the diagonals of U would give eigenvalues. However, the eigenvalues of U would be different from the eigen values of A because elementary row operations, E, change A. You would have to do EAx = lambad*Ex
Just watched christopher nolan movie... Super lecture..
for the equation: (A- lanbda* I)* X = 0, why should (A- lambda*I) needs to be singular?
I think otherwise the solution would be X be zero vector. and if (A-lambda*I) is singular, the whole combination can make some non-zero x to 0
We are interested in non trivial solutions, x to (A - lambda*I)*x = 0. For nxn matrix, if A is non-singular, the null space only contains the zero vectors (we are not interested in the zero space). Therefore, (A - lambda*I) needs to be singular to have non zero vectors, x in the null space
God bless you, Professor Strang.
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)
Good mathematics teachers are a gift from the gods.
I want to confirm, if we can find the eigen vectors for Q=[[0 -1]
[1 0]]
If yes, please help with its solution.
Thank you
When you form the characteristic equation Q-lambda * I
You get lambda**2 + 1 = 0
Which has complex roots, hence for a real vector space, we can’t find the eigenvalues of this matrix
How does this idea transfer to eigenvalues and eigenfunctions of differential operators? They don't quite look like matrices, do they?
But what if the first Eigen value found was wrong, based on addition and equality the othe will also be wrong
Jackpot unearthed....blessed
The next lecture can be....pure happiness. :D
What would the corresponding eigenvectors for -i and I at 41:00. Ther null space for the matrix is zero so are they the zero vector for both
i think they are [-i 1] and [i 1]
why at the last example there is no second eigen vectors .. why it can't be equal {-1 , 0 } .. is that because it will be // to it in the other direction .. and there is no one perpendicular to it ??
{-1,0} can be expressed as -1{1,0} we are looking for linearly independent vector
Amazing lecture. Thanks from Italy
Eigen in Thamizh I Head kann eye how eyes fit into head properly ie eigen vector and eigen value.
39:02 What is that with the subs??
Wow, that is weird. Good catch though! We're working on fixing it.
@@mitocw 👍
Best explanation ever I have seen
Basic concepts of streamlining automation. Conveyors systems.
Dr. Strang: in the example where you draw that plane in 3space does that plane contain the origin?
I think you might be thinking of a Euclidean space. A vector space does not have an origin. It's the set of tangent vectors (translations) of the elements (points) of a Euclidean space (not a mathematician, just trying to remember the basics here). It is much better to learn to think about vectors as translation (and in some cases as rotations) than to pretend that we are in a space with an origin. That's the actual physical meaning of vectors.
all vectors spaces needs to have the zero vector. This is because to be a vector space, it must be closed under addition and multiplication of scalars. So we have to be able to multiply by zero and still be in the vector space