Gilbert Strang is a towering testimony to why superb teaching is much more important to learning than digital pyrotechnics. His conforting humble stuttering shows us that he still today is in awe by this formidable piece of mathematics and invites us to recognize and confront our own difficulties in learning. Thak you professor! I admire you from afar with great joy and personal enrichment.
I"m in my first linear algebra course and am in awe of how immensely powerful this branch of mathematics is. MIT is fortunate to have a superb math teacher like Prof. Strang.
My Physics professors: *exhales in an annoyed fashion* "I really couldn't care less about the fact that I skipped 3 steps in my work while explaining a new concept this is extremely obvious and if you can't see it, I don't know how you made it into this class." Gilbert Strang: "I did a matrix multiplication I didn't prepare you for. I'm really sorry." Mr Strang I would literally die for you.
Much better than my prof who always tries to explain some very simple concepts in the most complicated fancy way so that it might make him look more qualified. The best prof should explain complicated concepts in the easiest and most comprehensible manner as possible
You are just a genius Gilbert! This is why you teach at MIT and wants to throw light on the shadows of ignorance in education round the globe. I am in bliss Sweet Angel!
I just love Prof.Dr.Strang's passion for teaching. He is such an amazing teacher. Having searched a lot of places to get an intuition about how different or same are eigen value decomposition and diagonalization of a matrix, voila, found all in one place. So glad to be learning concepts directly from a great mathematician like him.
So the column space of A or "transformed space by A" is the span of its eigenvectors! This makes sense of so many things you're the best Linear Algebra guy ever you legend
First of all I would like to thank you sir for share your knowledge freely!I think it's wonderful for everyone who learn Multivariate analysis course....He/She must watch your videos.....Please share more of Calculus & other branch of mathematics...
A^n = V * L^n * V^(-1) is actually eigenvalue decomposition of n-th power of A. Mr. Strang's illustration on how taking powers && taking differentials are like moving discretely && continuously are very a novel idea to me
There is also the notion of simultaneous diagonalization, meaning two diagonalizable matrices A and B consist of a basis of vectors which are both eigenvectors of A and B at the same time. Given diagonalizable matrices A and B, the subset of all diagonalizable matrices C which are simult. diag.able with A and B with the same base change matrix, they actually form a subspace of Mat_nxn(K) (the vector space of nxn square matrices over the field K)! And since A and B are obviously simultaneously diagonalizable with themselves, we know (for A=/=0 or B =/=0 matrix) that this subspace is not just the zero subspace. Furthermore, multiplying two matrices which are simultaneously diagonalizable yields a matrix which is again diagable with the same eigenvectors as basis of vector space, and the eigenvalues are just λ1μ1, λ2μ2, …, λ_n*μ_n. And also adding them keeps them simult. diagable. One can also show commutativity under matrix addition and multiplication, anf left and right distributivity is given. Right now these form a commutative ring (since for every C, also -C is inside, 0 and 1 are also inside and unique). If we now let A and B be invertible, all simultaneously diagonalizable matrices with A and B are also invertible (except 0). Since now every matrix in this subset except the zero matrix has a multiplicative inverse, we get a new field! This field is embedded in the field of all invertible matrices which commute with A and B(but I don‘t know if these are the same or not)
This of course works only if V is a square matrix and non-singular; otherwise, inverse V does not exist and the entire technique crashes. On the other hand, the SVD decomposition works for all matrices even those that are singular, because the method incorporates the transpose in place of the inverse.
Each time you operate the same matrix on an eigenvector, you get back the same vector, just multiplied by its eigen value. So it's rather obvious that any n-th power of any matrix will have the same Eigen vectors, and Eigen values just get raised to the n-th power!
You have no idea how incredibly helpful those short little pauses to backtrack a little and clear things up are. Thank you.
Thank youtube
Gilbert Strang is a towering testimony to why superb teaching is much more important to learning than digital pyrotechnics. His conforting humble stuttering shows us that he still today is in awe by this formidable piece of mathematics and invites us to recognize and confront our own difficulties in learning. Thak you professor! I admire you from afar with great joy and personal enrichment.
I"m in my first linear algebra course and am in awe of how immensely powerful this branch of mathematics is. MIT is fortunate to have a superb math teacher like Prof. Strang.
MIT is lucky to have such a great lecturer.
Prof Strang is the best in Linear Algebra.
1.5x speed + Gilbert Strang = happiness
gotta appreciate how he said "I did that without preparing you for it", that was so humble.
My Physics professors: *exhales in an annoyed fashion* "I really couldn't care less about the fact that I skipped 3 steps in my work while explaining a new concept this is extremely obvious and if you can't see it, I don't know how you made it into this class."
Gilbert Strang: "I did a matrix multiplication I didn't prepare you for. I'm really sorry."
Mr Strang I would literally die for you.
Much better than my prof who always tries to explain some very simple concepts in the most complicated fancy way so that it might make him look more qualified. The best prof should explain complicated concepts in the easiest and most comprehensible manner as possible
These mathematical tools are very important in science and engineering. Dr. Strang is an incredible human being for linear algebra.
Thank you MIT OCW! Prof. Strang is the ultimate contributor to education! Thank you!!
You are just a genius Gilbert! This is why you teach at MIT and wants to throw light on the shadows of ignorance in education round the globe. I am in bliss Sweet Angel!
A Life time asset ❤ priceless gift by The sir Gilbert Strang
I just love Prof.Dr.Strang's passion for teaching. He is such an amazing teacher.
Having searched a lot of places to get an intuition about how different or same are eigen value decomposition and diagonalization of a matrix, voila, found all in one place. So glad to be learning concepts directly from a great mathematician like him.
Tushara Devi again, Indians are everywhere 😀
This still remains to be the best video explaining this stuff!
Thank heavens for this kind man :) More professors need to post high quality videos like this! This is super helpful! Thank you MIT!
THE BEST AND MOST PASSIONATE CLASSES I HAVE EVER WATCHED ON THE TOPIC
Prof Gilbert Strang, thank you for the explanation. I bow to you _/\_
Aditya Gaykar I’m on my knees
I wish the professors at my university were this easy to understand!
Amazingly succinct and powerful - so much important stuff in just 10 minutes. Thanks prof strang.
what an absolute joy of sitting through a course taught by prof. strang.
You're a great lecturer! :)
Thank goodness for videos like these.
I suspect it helps that the lectures are aimed at engineers, rather than at mathematicians. For whatever reason, they are certainly wonderful.
So the column space of A or "transformed space by A" is the span of its eigenvectors! This makes sense of so many things you're the best Linear Algebra guy ever you legend
Professor Gilbert Strang is the Stronkest at Linear Algebra! He is Lord King Captain General Warlord Supreme Commander of Linear Algebra!!!! Stronk!
Came here to learn why diagonalizing a Hamiltonian is important and learnt from a real teacher!
Thank you so much , excellent video.The best teacher that I ' ve seen until now.
"That's very nice... that's very nice..."
I just had this in my lacture but didnt quite understand where the diagonal matrix came from but this cleared it up for me, thank you professor
literally THE BEST TEACHER...
Boss of Linear Algebra
First of all I would like to thank you sir for share your knowledge freely!I think it's wonderful for everyone who learn Multivariate analysis course....He/She must watch your videos.....Please share more of Calculus & other branch of mathematics...
Does V inverse always exist?
A^n = V * L^n * V^(-1) is actually eigenvalue decomposition of n-th power of A. Mr. Strang's illustration on how taking powers && taking differentials are like moving discretely && continuously are very a novel idea to me
Dear Prof, You are a fantastic teacher. Thank you very much.
You are the best Prof Strang!Thank you!
a very good teacher.
More than 80 years old, but taught better than the faculty of most Math schools in the world.
I can't believe that he can make this problem so easy for me to understand! Thx
I just love you Professor.
I let me go express my felling that you are the best Pr I have Seen.
I just love the lectures. You are the best sir. Kudos to you.
Prof. Strang is AWESOME
There is also the notion of simultaneous diagonalization, meaning two diagonalizable matrices A and B consist of a basis of vectors which are both eigenvectors of A and B at the same time. Given diagonalizable matrices A and B, the subset of all diagonalizable matrices C which are simult. diag.able with A and B with the same base change matrix, they actually form a subspace of Mat_nxn(K) (the vector space of nxn square matrices over the field K)!
And since A and B are obviously simultaneously diagonalizable with themselves, we know (for A=/=0 or B =/=0 matrix) that this subspace is not just the zero subspace.
Furthermore, multiplying two matrices which are simultaneously diagonalizable yields a matrix which is again diagable with the same eigenvectors as basis of vector space, and the eigenvalues are just λ1μ1, λ2μ2, …, λ_n*μ_n.
And also adding them keeps them simult. diagable.
One can also show commutativity under matrix addition and multiplication, anf left and right distributivity is given. Right now these form a commutative ring (since for every C, also -C is inside, 0 and 1 are also inside and unique). If we now let A and B be invertible, all simultaneously diagonalizable matrices with A and B are also invertible (except 0).
Since now every matrix in this subset except the zero matrix has a multiplicative inverse, we get a new field!
This field is embedded in the field of all invertible matrices which commute with A and B(but I don‘t know if these are the same or not)
we appriciate MIT and youtube for giving us our brain food
thanks proff gilbert strang
we also have herb gross for calculus
"Eye"-gen vectors and "eye"-gen values.
Thank you Dr. Strang, great video indeed
And this particular video was exceptionally helpful to me. Thank you!
Reviewing for my final. Thank you so much for making it so easy.
the best maths teacher in the universe including the ultragenius aliens in the space
that is a really absolutely wonderful video!!Thank you very much
You are such a wonderful teacher!
Understood very clearly, thank you very much! :)
love this prof.
makasih eyang strang :) jadi enak dan simple kalo bapak yang ngajar
What a great teacher!
simply great
This guy is incredible
he makes linear algebra so beautiful to me
Thank you, very helpful explanation.
both strang and mathematics are really cute
This vid has made my life!
Salute to you from Japan
Super helpful and thank you so much
If only all professors were half as good as Professor Strang.
Thank you so so much sir.
such a great explanations.
Gilbert is a good guy.....
11:19 Professor Strang gave us the secret to time travel
If time travel was possible where are our guests from the future
This is beautiful!
This is beautiful...
This of course works only if V is a square matrix and non-singular; otherwise, inverse V does not exist and the entire technique crashes. On the other hand, the SVD decomposition works for all matrices even those that are singular, because the method incorporates the transpose in place of the inverse.
Thank you.
Oh damn, You enlightened me. Thank you very much!
great lecture thank you
What a great mathematician!
thanks gil
only the rocking star of linear algebra can do this
Thank you thank you
Thank you
legend ,most of the tutorial didnt say the whole thing ,they just use the definition.
OH so clear!! Thanks a lot!
GOATbert Strang
he is a legend.....
till 18-03-21
I was remembered that formula..........
GOD real GOD
Naice
Thanku MIT
beautiful
cleared a lot of doubt❤
Thank you sir
Each time you operate the same matrix on an eigenvector, you get back the same vector, just multiplied by its eigen value. So it's rather obvious that any n-th power of any matrix will have the same Eigen vectors, and Eigen values just get raised to the n-th power!
4:20 How can V have inverse? Isn't it a non square matrix?
Real Pro !
So for any N X N matrix do we always have N eigenvalues and eigenvectors?
"now that I have it in a matrix form here I can mess around with it." lol in lib
This video makes me wish RUclips had a superlike! 😅
It is so helpful.
Professor 🙏Love from india
this is trippy
16 people still exponentiate their matrices by multiplying them by itself