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.
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.
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
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!
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.
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You have no idea how incredibly helpful those short little pauses to backtrack a little and clear things up are. Thank you.
Thank youtube
Prof Strang is the best in Linear Algebra.
MIT is lucky to have such a great lecturer.
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.
gotta appreciate how he said "I did that without preparing you for it", that was so humble.
These mathematical tools are very important in science and engineering. Dr. Strang is an incredible human being for linear algebra.
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.
Tushara Devi again, Indians are everywhere 😀
1.5x speed + Gilbert Strang = happiness
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
Thank you MIT OCW! Prof. Strang is the ultimate contributor to education! Thank you!!
A Life time asset ❤ priceless gift by The sir Gilbert Strang
This still remains to be the best video explaining this stuff!
what an absolute joy of sitting through a course taught by prof. strang.
I wish the professors at my university were this easy to understand!
Came here to learn why diagonalizing a Hamiltonian is important and learnt from a real teacher!
Prof Gilbert Strang, thank you for the explanation. I bow to you _/\_
Aditya Gaykar I’m on my knees
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
THE BEST AND MOST PASSIONATE CLASSES I HAVE EVER WATCHED ON THE TOPIC
Thank heavens for this kind man :) More professors need to post high quality videos like this! This is super helpful! Thank you MIT!
Amazingly succinct and powerful - so much important stuff in just 10 minutes. Thanks prof strang.
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
Professor Gilbert Strang is the Stronkest at Linear Algebra! He is Lord King Captain General Warlord Supreme Commander of Linear Algebra!!!! Stronk!
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...
I suspect it helps that the lectures are aimed at engineers, rather than at mathematicians. For whatever reason, they are certainly wonderful.
literally THE BEST TEACHER...
More than 80 years old, but taught better than the faculty of most Math schools in the world.
Thank goodness for videos like these.
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.
Thank you so much , excellent video.The best teacher that I ' ve seen until now.
I can't believe that he can make this problem so easy for me to understand! Thx
You're a great lecturer! :)
we appriciate MIT and youtube for giving us our brain food
thanks proff gilbert strang
we also have herb gross for calculus
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)
Boss of Linear Algebra
I let me go express my felling that you are the best Pr I have Seen.
Does V inverse always exist?
You are the best Prof Strang!Thank you!
Reviewing for my final. Thank you so much for making it so easy.
Prof. Strang is AWESOME
a very good teacher.
"Eye"-gen vectors and "eye"-gen values.
I just love you Professor.
he is a legend.....
till 18-03-21
I was remembered that formula..........
GOD real GOD
I just love the lectures. You are the best sir. Kudos to you.
Thank you Dr. Strang, great video indeed
he makes linear algebra so beautiful to me
makasih eyang strang :) jadi enak dan simple kalo bapak yang ngajar
And this particular video was exceptionally helpful to me. Thank you!
that is a really absolutely wonderful video!!Thank you very much
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.
"That's very nice... that's very nice..."
This guy is incredible
both strang and mathematics are really cute
If only all professors were half as good as Professor Strang.
Salute to you from Japan
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!
What a great mathematician!
Gilbert is a good guy.....
simply great
You are such a wonderful teacher!
Super helpful and thank you so much
the best maths teacher in the universe including the ultragenius aliens in the space
This vid has made my life!
This is beautiful...
Thank you, very helpful explanation.
Understood very clearly, thank you very much! :)
This video makes me wish TH-cam had a superlike! 😅
What a great teacher!
love this prof.
only the rocking star of linear algebra can do this
he writes z just like my parents! shout out from really far away thankkkkkkks
Thank you.
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.
Thank you so so much sir.
This is beautiful!
Oh damn, You enlightened me. Thank you very much!
legend ,most of the tutorial didnt say the whole thing ,they just use the definition.
such a great explanations.
thanks gil
cleared a lot of doubt❤
11:19 Professor Strang gave us the secret to time travel
If time travel was possible where are our guests from the future
Thank you thank you
Thank you
16 people still exponentiate their matrices by multiplying them by itself
Thanku MIT
great lecture thank you
beautiful
OH so clear!! Thanks a lot!
GOATbert Strang
this is trippy
Professor 🙏Love from india
Thank you sir
Real Pro !
Naice
It is so helpful.
Thanks Lot sir
thanks
thank. it help a lot