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Matrix Multiplication

matrixmultiplication.xyz

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Re: Matrix Multiplication

#81

Post by ducttapecrown »

pash wrote: post #76
nilkn wrote: post #64

Earlier quoted context omitted.

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 introd…

Axler's book is very good. An alternative to consider is Paul Halmos's much older Finite-Dimensional Vector Spaces [0]. Both books take the same basic approach, and the proofs of the major theorems are substantially the same. (Halmos's book is well known and well liked and was probably Axler's starting point.) Axler's book covers more ground (most notably, Halmos presents the polar decomposition but not the singular-…

Axler's book was free for me from my university's SpringerLink thingy.

Re: Matrix Multiplication

#82

Post by jayajay »

It's easier to think of matrix multiplication by computing the matrix elements, which means reducing the problem to NM vector dot products.

To get the element of the result matrix at position (n,m), compute the inner product , where r_n is the nth row vector of the left matrix and c_m is the mth column vector of the right matrix. Once you try it, you'll see that it's also much easier to visualize than this strange, unintuitive approach.

With great power comes great responsibility. That domain name belongs to that person. I hope they do society a favor and put more content regarding MM that that, because they chose one of the least intuitive ways of thinking about it. Put more ways up, or ditch the domain.

If the goal is to get people to understand what matrices are, the best way is to teach them about operators, transformations, vector spaces and linear algebra in general, because this is really the only way to fully understand what's going on without relying on some heuristic.

If the goal is to get people to remember how to do matrix multiplication, at least put up more ways of doing it.

Re: Matrix Multiplication

#85

Post by romwell »

staltz wrote: post #69
user1713952 wrote: post #62

Earlier quoted context omitted.

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 a…

To back up the parent, unfortunately, the audience is not always qualified to judge the quality of your work - for the simple reason that they have no idea what they are missing!

What they are missing from your explanation is "why". Your animation shows how to multiply two matrices, and anyone confused about the mechanics of doing that might find it helpful.

But the actual mechanics are not of much use if you don't know why they are this way. Everything in mathematics is made up by humans, so why would someone at some point write down this particular set of rules to apply to rectangular arrays of numbers?

To that end, I reaffirm the parent's opinion that your visualization only helps to confuse people, giving them a false sense of understanding. There is already too much emphasis in education on "how", and too little on "why"; the comments from the teachers you mentioned are downright scary to me.

Perhaps you will take this comment as yet another critique that is too harsh - but it must be done, so that others don't mistake your animation for an explanation of concepts behind the operation. When one understands the concepts, visual mnemonics - like the one you created - are not needed.

To start a productive discussion, however, (which, hopefully, might lead to you using your skills to come up with a version 2 of your work), I am very curious to hear your answers to these questions - which people confused about matrices might have:

1. What information is actually stored in a matrix? Why write down a bunch of numbers in a rectangle instead of, say, a list or a triangle (like Pascal's triangle)?

2. Why would one want to "multiply" two matrices? That is, why would one want to combine two matrices, and what information would the "product" matrix store?

3. Why do the rules for "multiplication" actually yield the result desired in (2)? Wouldn't some simpler rule do the same? For instance, if we have two 2x2 matrices, why not just multiply the numbers there elementwise (top left by top left, top right by top right, etc.)?

My claim is that anyone who can answer these questions wouldn't need your visualization - and, conversely, your visualization doesn't help answer these questions at all.

I'd be glad to be proven wrong, so I'm looking forward to your explanations of the three questions above.

Re: Matrix Multiplication

#86

Post by tvural »

Matrices are linear maps, and matrix multiplication composes the linear maps. Now statements like the determinant of the product is the product of the determinants, the trace is preserved under similarity transforms (since similarity transforms just rewrite the linear map in a different basis), etc. are intuitive.

I've always felt that these explicit calculations don't really get to the point. You can memorize them and still not really understand what's going on.

Re: Matrix Multiplication

#87

Post by staltz »

romwell wrote: post #85
staltz wrote: post #69

Earlier quoted context omitted.

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 a…

To back up the parent, unfortunately, the audience is not always qualified to judge the quality of your work - for the simple reason that they have no idea what they are missing! What they are missing from your explanation is "why". Your animation shows how to multiply two matrices, and anyone confused about the mechanics of doing that might find it helpful. But the actual mechanics are not of much use if you don't k…

I totally acknowledge that it is missing the "why", I knew that from the beginning. The widget I built simply answers the "how". It's a very focused tool, and this kind of tool may be very useful in itself. It's a tool to be used in the context of other tools, like books and courses. Your ordinary calculator never answers the why question. And if the audience, who includes both students and teachers, isn't qualified to judge the quality of the work, who exactly is?

On a side note, it's disgusting (I find that an appropriate word) how hackernews threads are so harsh and negative. You can do a fun exploration: search for historical threads announcing tools and services that were revolutionary (Angular.js or React.js, to name a few), and you'll find a plethora of negative comments. If hackernews discussions are good judges of quality, absolutely everything in this world sucks.

Re: Matrix Multiplication

#88

Post by staltz »

jayajay wrote: post #82

It's easier to think of matrix multiplication by computing the matrix elements, which means reducing the problem to NM vector dot products. To get the element of the result matrix at position (n,m), compute the inner product , where r_n is the nth row vector of the left matrix and c_m is the mth column vector of the right matrix. Once you try it, you'll see that it's also much easier to visualize than this strange, u…

Nope, won't ditch the domain. I'd recommend you ditch that type of speech.

Re: Matrix Multiplication

#90

Post by leshow »

romwell wrote: post #85
staltz wrote: post #69

Earlier quoted context omitted.

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 a…

To back up the parent, unfortunately, the audience is not always qualified to judge the quality of your work - for the simple reason that they have no idea what they are missing! What they are missing from your explanation is "why". Your animation shows how to multiply two matrices, and anyone confused about the mechanics of doing that might find it helpful. But the actual mechanics are not of much use if you don't k…

I don't know how you think that school systems, especially in the west, place more emphasis on "how" than "why". Western schools most definitely place their emphasis on "understanding" rather than "doing". I think that all this does it create a false sense of understanding. Students think they understand why something is done, then when they go to do an actual problem they are totally lost.

I think this is a big reason why american students rate themselves as being very confident in their understanding, but performing rather poorly compared to other nations. I'll have to look for the citation to back that up.

Just my 2cents.

edit: http://www.newgeography.com/content/003840-high-confidence-n...

there were other more recent surveys on this topic, this is all i can find right now

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