I really like the transparent whiteboard, you're like a magic caster starting a magic spell in some anime. Also really good series, clear explanations and easy to follow. Even for someone like me who has limited knowledge of linear algebra.
When your vector space is C^n you use something called the Hermitian adjoint instead of the transpose. The really cool thing is that the Hermitian adjoint collapses to the transpose when the imaginary component vanishes, like you mentioned. Loved the analogy with star constellations, that's a keeper!
I was wondering why most teachers who use a giant glass board as a blackboard are left-handed, then I saw you and thought great! found a right-handed one, then realized you are indeed left-handed, and all others are right-handed => They actually flip the video at the end, so your writings face the correct direction according to us! (: Anyway, great lecture. Helped me a big deal today. Thank you so much!
Enjoying the series to the optimum level! Kudos to Professor Steve and his teaching style. By the way, I seem to not understand something: How is it a "sphere" (3D) that contains the unit vectors which belong to the space of m dimension? Probably I am not understanding something, any help will be appreciated. Thanks!
He explicitly said m-dimension sphere, but I think the key was "a body of all possible unity vectors". Drawing and calling it a sphere I believe was only to make use of the 3D-visuability of the mind.
This is a really good question. A lot of physics problems are naturally written using complex numbers (Schrodinger's equations, the motion of a mass on a spring, or really any oscillator), and then the physics equations involve complex functions. When we linearize these equations, we often get complex matrices. The SVD of these linear operators will tell us what inputs will get amplified, and what outputs will be the largest. This is called resolvent analysis.
I am binge watching your lectures. I have a question though:- If circle is transformed into ellipse by matrix multiplication, then shouldn't be possible to represent that matrix as product of diagonal matrix and rotation matrix? what's the need for three matrices in SVD? I am assuming the parametric eqn of ellipse is x=acos(theta) and y=bsin(theta)
if I do the decomposition in R, U^t * U gives the identity matrix but U*U^t doesnt. The matrices U and V are not even square. Does it compute by default the 'econ' variant ?
What I never liked in Linear algebra is the nuances of the terms orthogonal and unitary. And orthogonal matrices are defined as strictly real. My favourite examples are A = [0 i; -i 0] and B = [sqrt(2) i; i -sqrt(2)] If you multiply A by A^* you get I so it's unitary but not orthogonal cause A * A^T is not I (and it could not be orthogonal cause it has complex elements). Matrix B if multiplied by B^T will give I but its also not technically orthogonal cause it has imaginary elements, and it's not unitary also cause B times B^* is NOT I.
@@elenichatzikyriakou2721 Yes, he's definitely a leftie. The whole image is mirrored after the fact, otherwise all the text would be reversed for us viewers. His ring is on his left hand, which is the norm in the US.
![Unitary Transformations and the SVD [Matlab]](http://i.ytimg.com/vi/_wOt50VnJw4/mqdefault.jpg)
![Unitary Transformations and the SVD [Matlab]](/img/tr.png)
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/n.gif)
These videos coupled with the book really take learning to the next level. Incredible!
exactly. thats what im doing too
Just ordered the book! Support those that give you value.
I really like the transparent whiteboard, you're like a magic caster starting a magic spell in some anime.
Also really good series, clear explanations and easy to follow. Even for someone like me who has limited knowledge of linear algebra.
These are one of the best videos I watch on SVD. Thanks, professor Steve
Thank you Mrs. Brunton! God bless you forever
When your vector space is C^n you use something called the Hermitian adjoint instead of the transpose. The really cool thing is that the Hermitian adjoint collapses to the transpose when the imaginary component vanishes, like you mentioned. Loved the analogy with star constellations, that's a keeper!
I was wondering why most teachers who use a giant glass board as a blackboard are left-handed, then I saw you and thought great! found a right-handed one, then realized you are indeed left-handed, and all others are right-handed => They actually flip the video at the end, so your writings face the correct direction according to us! (:
Anyway, great lecture. Helped me a big deal today.
Thank you so much!
lolz
Flip the video otherwise he has to write in reverse like they do in the Navy Combat Information Center. It also a matrix transformation!!
In my opinion, at 11:34, a 3d sphere multiply 3x2 matrix (not 2x3 matrix) to get a 2d ellipsoid
Good question. In this example I am right multiplying by X, so I think the dimensions are correct.
I thought the same! The transparent screen switches left and right so that is why we were confused
Steve Brunton? More like "Super Tim Burton", because these lectures are scary good!
Whoa! Brain is getting reconfigured, developing new folds.........
Enjoying the series to the optimum level! Kudos to Professor Steve and his teaching style. By the way, I seem to not understand something: How is it a "sphere" (3D) that contains the unit vectors which belong to the space of m dimension? Probably I am not understanding something, any help will be appreciated. Thanks!
He explicitly said m-dimension sphere, but I think the key was "a body of all possible unity vectors". Drawing and calling it a sphere I believe was only to make use of the 3D-visuability of the mind.
he just employed a 3d-sphere to illustrate a hypersphere in R^m.
Is a matrix transformation of a sphere necessarily an ellipsoid?
What's the significance of having complex data for X? What's an application of why the data would be complex? TIA.
This is a really good question. A lot of physics problems are naturally written using complex numbers (Schrodinger's equations, the motion of a mass on a spring, or really any oscillator), and then the physics equations involve complex functions. When we linearize these equations, we often get complex matrices. The SVD of these linear operators will tell us what inputs will get amplified, and what outputs will be the largest. This is called resolvent analysis.
@@Eigensteve Oh wow, that's super interesting; thanks for the answer 😁
I am binge watching your lectures. I have a question though:-
If circle is transformed into ellipse by matrix multiplication, then shouldn't be possible to represent that matrix as product of diagonal matrix and rotation matrix? what's the need for three matrices in SVD? I am assuming the parametric eqn of ellipse is x=acos(theta) and y=bsin(theta)
if I do the decomposition in R, U^t * U gives the identity matrix but U*U^t doesnt. The matrices U and V are not even square. Does it compute by default the 'econ' variant ?
Would it be possible to cover dual spaces?
Does the transformation from a sphwrek has to be always ellipsoid .
Professor I want follow your linear algebra lectures, do you think poeple graduate bachelor degree in mathematic can follow these lectures?
What I never liked in Linear algebra is the nuances of the terms orthogonal and unitary. And orthogonal matrices are defined as strictly real. My favourite examples are A = [0 i; -i 0] and
B = [sqrt(2) i; i -sqrt(2)]
If you multiply A by A^* you get I so it's unitary but not orthogonal cause A * A^T is not I (and it could not be orthogonal cause it has complex elements).
Matrix B if multiplied by B^T will give I but its also not technically orthogonal cause it has imaginary elements, and it's not unitary also cause B times B^* is NOT I.
i LOVE this series, but oof, that marker squeak!
skip two math classes, come back to this.
shouldn't v lie in R^n and u in R^m ?
👍🏼
Holy s$&@! that was cool
The author teaches very well, but what makes me surprised is how he writes the reverse word! this impossible!
تمام الخيال الطريفي لكل خير وصحة وسلامة وكامل عالم أجمعين قووووو
He looks like harrison wells
about to comment the same
3:07 As we travel through the "Celestial Sphere", not the solar system hhhh
A fellow leftie
Is it? Because I rarely see rings on the left hand, but it would definitely explain writing in 'opposite'.
@@elenichatzikyriakou2721 Yes, he's definitely a leftie. The whole image is mirrored after the fact, otherwise all the text would be reversed for us viewers. His ring is on his left hand, which is the norm in the US.