Visualize Different Matrices part2 | SEE Matrix, Chapter 1
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- Опубліковано 6 чер 2024
- Visualizing, identity matrix, scalar matrix, reflection matrix, diagonal matrix, zero matrix, shear matrix, orthogonal matrix, projection matrix, inverse of a matrix.
Chapters:
0:00 Shear Matrix
1:57 Orthogonal Matrix
5:20 Projection Matrix
7:30 Inverse
9:35 What exactly is a Matrix ?
Hopefully providing more intuition about matrix transformation on vectors and making the very abstract object of matrix, more relatable to us. This visual approach to matrix transformation also is the foundation to the Grand Finale of visualizing SVD.
This video wouldn’t be possible without the inspiration of the legendary 3b1b :
/ 3blue1brown
and the animation software - Manim, which he wrote:
/ 3blue1brown
and the Manim Community:
docs.manim.community/en/stabl...
Video Sins:
1. It’s clear we have 4 different ways of shear transformation in 2D. But in 3D, are there 6 different ways ? Or are there infinite number of ways ? When I tried to look up the exact definition of shear transformation in 3D, I actually couldn’t find a very rigorous definition. Base on this definition: www.cut-the-knot.org/WhatIs/W..., it seems like there should be more than 6 different ways. I feel like shear transformation is one of those things that doesn’t actually doesn’t need a formal definition, it’s not like symmetric or orthogonal or identity matrix, which serves significance in other proofs of linear algebra. I could be wrong tho.
2. 3:38. “Orthogonal matrix always produces a rotational transformation to some degree” (firstly, pun intended). But here is the war on definition again. What exactly is a rotation ? The main debate is if the definition of rotation should also generalize improper rotation en.wikipedia.org/wiki/Imprope..., which happens when the determinant of the orthogonal matrix is -1. To be honest, I have no idea. Perhaps the better way I should phrased it was “a rotation transformation can always be described an orthogonal matrix” rather the other way around.
Image Credits:
Mugen (samurai champloo), Spike (cowboy bebop), Dandy (space dandy), Obito (naruto), John Green, Gilber Strang, Eren (attack on titan)
Music Credits to Nujabes, may he rest in beats:
1. beat laments the world
2. aruarian dance
3. flowers
4. modal soul
Other Music:
1. Going Down by Jake Chudnow
2. *MACINTOSH PLUS - リサフランク420 / 現代のコンピュー*
My Email: vizkernel@gmail.com
Feel free to send me suggestion for future video ideas.
This channel is criminally underrated, watching the video i was baffled by how well explained and how WELL EDITED THIS IS, I cannot fathom the amount of work that went into this especially the visualisations ! I thought this channel had millions of subscribers and i was wondering how i missed it, only 9K ?!!! I'm sorry this is outrageous.
PS: I think 3Blue1Brown would definitely be proud.
I asked ChatGpt and it could not compete with this guy's videos. This is Gold. Whoever did these videos should share their name. A great honour beholds them. 🎉❤
In these 2 parts I could say I've learnt more about matrices than (sadly) during the high school and even university courses on linear algebra. It gave me so much needed intuition behind all those theoretical concepts. I could only wish our educational system was so engaging and demonstrative like these videos! Thank you very much!
This channel deserves much more attention! ❤
Awesome, simply awesome. Only 51 comments for such a fabulous job is just not fair.
Looking forward to Chapter 2 !!
I like the character development of the potato!
Please do not stop doing videos, are just invaluable.
Keep doing these videos man, We just love them!
Phenomenal work! I'm very thankful to you for such a great content
These videos are invaluable! Thanks a lot. Please create more of such videos.
Man, these videos are gold!
Shared it with all my friends as a token of gratitude.
these are wonderful videos!
Incredible work!!
perfect.
Thank you for this fantastic video!
This is really great. You will get a better understanding at 3:39 with a unit circle.
awesome videos and I love the graphics, nice to see some of my favorite anime characters while studying linear algebra ;)
very clear. like it very much. Thanks.
brilliant videos, thanks a lot
This is so good.
Thank you so much love from india. You sorted out Lot of things
It's a great piece to summarize a hard topic. He has also made it very simple with the computer graphics. I've learnt linear algebra from Prof G. Strang and this just gives it life to explain those hard topics.
Amaizing content!!!!!!!!!!
sublime
Nujabes music in the back and Watanabe characters to describe. I like this channel!
nujabes in the background with anime characters to explain. 10/10 banger
Seriously, you're going from basic visualisation to spectral decomposition I'm the second vid?! Brave man.
I don't mind you skipping Gaussian elimination but i do think you should have shown what a linear combination is. Also independent and dependent bases.
Was a lovely video though and, like you, I prefer the geometric interpretation.
Remarkable videos! 💜 Sadly the visualizations are not applied in Mathemstics classes in most universities. I first tried to get through all definitions but studying something without using or seeing it irl is pointless. If not for your vids I would have probably ended up learning little to nothing at all. Please continue! 😊
Love the detail of the least squares normal equation when talking about the data matrix (10:46)😂😹
Thank you for your information. Thanks for telling us about Manim. I want to make videos to teach 7 years old kids math with animation.
Is that a reference to 二向箔 :)
Great great great
Nujabes = Chefs kiss.
Thank you for sharing your video. Wish you health and wealth.
These videos, and other visualization videos of a similar nature, are very helpful in understanding much of underlying concepts. One thing, however... is it possible to perform some spell checking within the visualizations themselves? Sometimes the misspellings diminish the impact of the video. Is spell checking within the visualizations difficult, or is it difficult to correct the visualizations after a misspelling is detected later in the production process? Anyway, good work.
0:42 you can always work out the equations and it will be quite apparent what is going on!
Your videos are brilliant, they should be 1K times more popular!
But why do you need jazz in the background, it just distracts? ).
Thanks, helped me a lot!
vsauce music was a nice touch!
Bravooooooooo
brilliant!!
for those confused why is sqrt(2)/2 a unit vector, cuz at least I thought it should be equal to 1, here's the proper explanation:
A unit vector is a vector with a magnitude of 1. we find magnitude using |v| =√(x^2 + y^2) or also known as the distance formula between points OR ALSO KNOWN AS THE PYTHAGORUS THEOREM-ish ("-ish" cuz Pythagoras theorem is for right-angle triangles but it's fine it works here as well cuz when finding distance from two points, we can just imagine the distance to be a hypotenuse of an imaginary right-angle triangle and then apply the formula using √(x^2 + y^2) where y is the 'rise' and x is the 'run') .
normally, |v| =√ (1^2 + 0^2) = 1 OR |v| =√ (0^2 + 1^2) = 1, hence they're a unit vector, but it's not always limited to that.
The expression √2/2 can represent the x or y component of a unit vector in two dimensions. For instance, normally if you have a unit vector pointing in the positive x-direction, its x-component would be 1 and its y-component would be 0. However, if you have a unit vector pointing at a 45-degree angle from the positive x-axis aka √2/2, both its x and y components would be √2/2, and the magnitude of the vector formed by these components would indeed be 1. This is because when you square and add the components, you get 1, then √1 is still 1, hence satisfying the condition of being a unit vector.
Sir, What is Geometric interpretation of Trace of a Matrix. Kindly make a video.
how you made a [ -sqrt(2)/2, sqrt(2)/2 ] into unit vector remains complete mystery to me....
i was stuck there too. can somebody explain this to me
@noitnettaattention @rhaegartargaryen0947 basically what he is saying is that for a vector ( column of the matrix) to be a unit vector then it needs to have a module of 1. The module of a vector is just like doing a Pythagoras theorem with x and y to find the hypothenuse. this said, generally people don't go that far and just use the following formula: module = sqrt(x^2 + y^2). the module of the vector is therefore sqrt(1/2+1/2) = sqrt(1) = 1 thus making it a unit vector.
this vector is very commonly seen on many applications since it has a simple 45º angle with the axis but any vector you can imagine that goes from the origin to a radius 1 circumference really qualifies as a unit vector (in R^2, 2 dimensions).
What he wants to explain with this is that there can be matrixes that have two unit vectors, but to have them orthogonal is yet another important and distinct characteristic that he shows some seconds after.
The good thing about the orthogonal unit vectors is that they are called "eigenvectors" and together with "eigenvalues" they will set a new set of axis for this transformation, as if the whole original axis is being rotated, as to have two perpendicular axis in a 2D space you can basically say that they are just some rotation or inversion of the original two axis.
Hope I helped!
Cheers
Orthonormal
For orthogonal matrices, I don’t believe that implies they have unit vectors for columns. Isn’t that reserved for orthonormal matrices?
Just want to correct my understanding it it is wrong, great video!
at 2:07 , orthogonal matrices need not have a unit column vectors, only orthonormal matrix, anyway which is a subset of orthogonal matrix, has unit column vectors. Am i right? If not, please enlighten me. Thank you for the awesome video.
"Unfortunately no one can be told what the Matrix is" - Morpheus
“- you have to see it for yourself.”
your content is amazing! but i feel you can probably rename your videos so they aren't discriminated by the algorithms!!!!
Nujabes ❤
I know this transformation, 3 bodies, reduce dimension 😁
7:01 Death's end reference?
It's not true that all orthogonal matrices represent rotations. Some orthogonal matrices are reflections. Specifically, orthogonal matrices with determinant = 1 are rotations, and orthogonal matrices with determinant = -1 are reflections.
A reflection is a rotation too , right ?
@@mb59621
I wouldn't say so. A reflection inverts the orientation of the space, hence the negative determinant, whereas a rotation does not.
Reflections are also their own inverse. Applying a reflection twice lands you back where you started, whereas it doesn't with most rotations.
Since orthogonal matrices are defined as those with an inverse equal to its transpose, this means that the transpose of a reflection matrix is equal to itself (since it is its own inverse). Therefore, reflection matrices are symmetric across their main diagonal (taking the transpose doesn't change the matrix), whereas most rotation matrices are not.
@@APaleDot good points to ponder while still learning the subject. At first I thought reflection is just a rotation about the normal , but ty for your reply , i found an inaccuracy in my intuition that a single rotation matrix built on this principle cannot reflect anything more than 1 point ! Which is what makes reflection matrices a bit unique.
@@mb59621 a reflection is a discrete transformation whereas rotation is the sum of infinitesimal transformations.
three body reference??/
How is root 2/2 , - root2/2 a unit vector
A unit vector is any vector with a length of 1. We can find the length of a vector using the Pythagorean theorem:
(length)² = x² + y²
= (sqrt(2) / 2)² + (-sqrt(2) / 2)²
= (2 / 4) + (2 / 4)
= 1
The square root of 1 is just 1, so we conclude that the length of the vector is 1, a unit vector.
nujabes + maths
二向箔哈哈哈