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Matrix Multiplication

matrixmultiplication.xyz

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Re: Matrix Multiplication

#61

Post by snovv_crash »

samuell wrote: post #35
daniel-levin wrote: post #19

This is a cool example of what Bret Victor calls an "Explorable Explanation" [0]. That said, I feel that it's more important to understand how and why matrix multiplication corresponds to a composition of linear transformations than learning the actual mechanics of doing the computation. You can get good at matrix multiplication without knowing what is going on. I view that as a less valuable activity than learning a…

For understanding the mechanics of matrix multiplication, I found it useful to think in an analogy consisting of a grid of two layers of pipes; One -- the input-pipes -- coming in one direction, and the other -- the output pipes -- laid in an orthogonal direction. Then there would be "taps" in the cross-sections between the input and output pipes, representing the numbers (multiplication factors, really) in the matri…

I really like that. And matrix * matrix would just have to be 3D, with a whole cube of taps. Sure, not easy to illustrate, but the concept extends well. It also makes it more obvious what the complexity of the operation is.

Re: Matrix Multiplication

#62

Post by user1713952 »

staltz wrote: post #48
user1713952 wrote: post #41

The idea is good but I am afraid it misses the point entirely. Multiplying both columns of the right hand side matrix to the left hand side matrix is really not helping the target audience understand what is going on. 2 operations are going on as the same time. I believe https://www.youtube.com/watch?v=XkY2DOUCWMU is a much better way to illustrate the matrix multiplication (and provide a clear explanation of the why…

Comments like these (aside from providing a nice link) just make authors of content sad. Why say such thing? You can say "here is another helpful link", but "misses the point entirely"? The point is to provide a self-evident mnemonic, as explained here: https://github.com/staltz/matrixmultiplication.xyz Not nice.

Mr Staltz, please allow me to elaborate. I did not mean to be mean at all, nor to reduce the value of your work, excuse me if I made you feel like that. I am actually grateful that you made this animation as it allows one to reflect on the concept.

If your kid proudly brings home a drawing, you are most probably going to greet him with warm encouragements no matter the quality of the art. The kid is in a process of developing his creative sense, learning, and forming his/her sense of reality. Comparing the drawing to what mature artists produce would certainly have a negative effect on the kid. The point is to provide a feedback relevant to the recipient of the critic.

The same goes here with this animation. I respect your work but obviously disagree with the usefulness of it. You made an animation to explain the matrix product, which is a really good idea. Alas, I think the realization of it gives a confused picture of what it is. I doubt that a student with no prior understanding of matrix products will get much enlightment. Too many operations are going on at the same time. This means that it misses its intended audience: students in need of building an intuitive understanding of the matrix product. That's what all those javascript (among others) animations (ex: http://jackschaedler.github.io/circles-sines-signals/dft_int...) are all about: make it simpler to grasp a concept. Unfortunately, though beautiful, the animation is not clear. At best it provides a visual mnemonic for multiplying matrices which maybe useful for some but not for the younger version of me in need of understanding :)

I realize that instead of providing such a harsh critic, I would have been better inspired to suggest improvements. Mea culpa :)

Re: Matrix Multiplication

#63

Post by vanderZwan »

ahane wrote: post #44
daniel-levin wrote: post #19

This is a cool example of what Bret Victor calls an "Explorable Explanation" [0]. That said, I feel that it's more important to understand how and why matrix multiplication corresponds to a composition of linear transformations than learning the actual mechanics of doing the computation. You can get good at matrix multiplication without knowing what is going on. I view that as a less valuable activity than learning a…

3Blue1Brown is an amazing youtube channel that provides very entertaining and educating visual explanation for all sorts of maths. He has a whole series on linear algebra, including matrix mulitplication: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

... and after a succesfull Patreon campaign[0], he's going to do "Essence of Calculus" next. I'm seriously excited!

[0] https://www.patreon.com/3blue1brown

Re: Matrix Multiplication

#64

Post by nilkn »

daniel-levin wrote: post #19

This is a cool example of what Bret Victor calls an "Explorable Explanation" [0]. That said, I feel that it's more important to understand how and why matrix multiplication corresponds to a composition of linear transformations than learning the actual mechanics of doing the computation. You can get good at matrix multiplication without knowing what is going on. I view that as a less valuable activity than learning a…

This is why I'm a big fan of Axler's Linear Algebra Done Right. The book's emphasis is on the concepts behind the calculations rather than the calculations.

I actually disagree with Axler on his avoidance of the determinant, though. I wish instead of avoiding it he'd spent more time developing it conceptually, as it's actually a fascinating construction. But to this day I have yet to find a gentler and better introduction to serious linear algebra than his book.

He recommends it as a second course, but I read it during my first course in the subject and considered it my "secret weapon". I truly believe that book is what allowed me to get a perfect score in the class -- I had a conceptual understanding that was just not possible to glean from the official course textbook.

Re: Matrix Multiplication

#65

Post by amelius »

How I would explain it:

- First, show how to multiply a row vector by a column vector.

- Then, show how to "simultaneously" multiply several row vectors by a single column vector by placing the row vectors in a matrix.

- Now, similarly, do the same with the column vector. And voila, matrix by matrix multiplication.

Re: Matrix Multiplication

#66

Post by justinpombrio »

snovv_crash wrote: post #61
samuell wrote: post #35

Earlier quoted context omitted.

For understanding the mechanics of matrix multiplication, I found it useful to think in an analogy consisting of a grid of two layers of pipes; One -- the input-pipes -- coming in one direction, and the other -- the output pipes -- laid in an orthogonal direction. Then there would be "taps" in the cross-sections between the input and output pipes, representing the numbers (multiplication factors, really) in the matri…

I really like that. And matrix * matrix would just have to be 3D, with a whole cube of taps. Sure, not easy to illustrate, but the concept extends well. It also makes it more obvious what the complexity of the operation is.

In a little more detail:

Look at the animation in the article, after the second matrix has been rotated and put on top of the first one. Then flip its top up so the two matrices are orthogonal. Finally, rotate the whole thing 90 degrees to the left (rotating along the axis that goes from the top of the page to the bottom of the page). Now you can see that the result will be a 2x3 matrix. And the values in each spot will be the sum of the products beneath it.

Re: Matrix Multiplication

#67

Post by maxiepoo »

FeepingCreature wrote: post #32

This is sounding increasingly like the Monad situation, where everybody has their own incompatible approach for explaining it, and none of them make any sense. I feel like I understand Matrices _less_ now than when I started reading around here.

If by "Monad situation" you mean trying to learn a concept by just reading a single blog post then sure. Trying to learn something by scattered blog posts without engaging in material is a bad idea no matter what the problem is. For some reason nobody expects to read a blog post and understand the fundamental theorem of calculus without first learning intuition and doing calculations with derivatives and integrals. On the other hand, if you have been working through a concept and you read a blog post that gives a different perspective than you're used to, it might help.

If you want to actually learn something, then get something like a book or series of posts and work through examples and explore the ideas yourself. But having more than one intuition is useful and complaining about the fact that there is more than one way to understand something is lazy.

Re: Matrix Multiplication

#68

Post by CalChris »

jperras wrote: post #53
t3nary wrote: post #23

Earlier quoted context omitted.

I feel very similar. Do you have any recommendations for resources to learn about why matrix multiplication is defined that way?

Axler's book, "Linear Algebra Done Right", is widely considered to be one of the best texts on linear algebra that focuses less on the mechanics and more on the structure of linear operators on vector spaces. http://linear.axler.net/

Yes, and the abridged version (no proofs, examples, and exercises) is downloadable for free. He avoids determinants altogether (well, until the 10th chapter).

http://linear.axler.net/LinearAbridged.html

Re: Matrix Multiplication

#69

Post by staltz »

user1713952 wrote: post #62
staltz wrote: post #48

Earlier quoted context omitted.

Comments like these (aside from providing a nice link) just make authors of content sad. Why say such thing? You can say "here is another helpful link", but "misses the point entirely"? The point is to provide a self-evident mnemonic, as explained here: https://github.com/staltz/matrixmultiplication.xyz Not nice.

Mr Staltz, please allow me to elaborate. I did not mean to be mean at all, nor to reduce the value of your work, excuse me if I made you feel like that. I am actually grateful that you made this animation as it allows one to reflect on the concept. If your kid proudly brings home a drawing, you are most probably going to greet him with warm encouragements no matter the quality of the art. The kid is in a process of d…

I think your elaboration just made it worse, as you compared my work to a kid's drawing, and assumed that the target audience will not benefit from that. I actually have proof against that. In Twitter, over 2000 people have retweeted, and countless people have commented along the lines of "I wish this was around when I was studying in school", or "now I finally understand it", and even school teachers saying thanks and how they'll use it in class next week. And, even this hackernews thread, for that matter.

Re: Matrix Multiplication

#70

Post by jakehoon »

This would have been so awesome to have in college. I stumbled upon RegExr the other day (http://regexr.com) which I absolutely love. It kind of shares this spirit of showing people how to do things that appear complex at first glance, but have a sort of addicting nuance to them. Thanks for sharing.
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