Is Matrix Multiplication Ugly?

Do you disagree with my take or think I’m missing Witt’s point? I’d be happy to hear from people who disagree with me.

Maybe the problem is that matrices are too general.

You can have very beautiful algorithms when you assume the matrices involved have a certain structure. You can even have that A*B == B*A, if A and B have a certain structure.

If the O(n^3) schoolbook multiplication were the best that could be done, then I'd totally agree that "it's simply the nature of matrices to have a bulky multiplication process". Yet there's a whole series of algorithms (from the Strassen algorithm onward) that use ever-more-clever ways to recursively batch things up and decrease the asymptotic complexity, most of which aren't remotely practical. And for all I know, it could go on forever down to O(n^(2+ε)). Overall, I hate not being able to get a straight answer for "how hard is it, really".

Ignore me then because I agree with you. :) He sounds like someone who upon first hearing jazz to complain it was ugly.

> sends the pair (x, y) to the pair (−x, y)

I know linear algebra, but this part seems profoundly unclear. What does "send" mean? Following with different examples in 2 by 2 notation only makes it worse. It seems like you're changing referents constantly.

Thanks for pointing this out. I’ll work on this passage tomorrow.

I think you're right that the inelegant part is how AI seems to just consist of endless loops of multiplication. I say this as a graphics programmer who realized years ago that all those beautiful images were just lots of MxNs, and AI takes this to a whole new level. When I was in college they told us most of computing resources were used doing Linear Programming. I wonder when that crossed over to graphics or AI (or some networking operation like SSL)?

Let me try.

In US schools during K-12, we generally learn functions in two ways:

1. 2-d line chart with an x-axis and y-axis, like temperature over time, history of stock price, etc. Classic independent variable is on the horizontal axis, dependent variable is on the vertical axis. And even people who forgotten almost all math can instantly understand the graphics displayed when they're watching CNBC or a TV weather report.

2. We also think of functions like little machines that do things for us. E.g., y = f(x) means that f() is like a black box. We give the black box input 'x'; then the black box f() returns output 'y'. (Obviously very relevant to the life of programmers.)

But one of 3blue-1brown's excellent videos finally showed me at least a few more ways of thinking of functions. This is where a function acts as a map from what "thing" to another thing (technically from Domain X to Co-Domain Y).

So if we think of NVIDIA stock price over time (Interpretation 1) as a graph, it's not just a picture that goes up and to the right. It's mapping each point in time on the x-axis to a price on the y-axis, sure! Let's use the example, x=November 21, 2025 maps to y=$178/share. Of course, interpretation 2 might say that the black box of the function takes in "November 21, 2025" as input and returns "$178" as output.

But what what I call Interpretation 3 does is that it maps from the domain of Time to the output Co-domain of NVDA Stock Price.

3. This is a 1D to 1D mapping. aka, both x and y are scalar values. In the language that jamespropp used, we send the value "November 21, 2025" to the value "$178".

But we need not restrict ourselves to a 1-dimensional input domain (time) and a 1-dimensional output domain (price).

We could map from a 2-d Domain X to another 2-d Co-Domain Y. For example X could be 2-d geographical coordinates. And Y could be 2-d wind vector.

So we would feed input of say location (5,4) as input. and our 2Dto2D function would output wind vector (North by 2mph, East by 7mph).

So we are "sending" input (5,4) in the first 2d plane to output (+2,+7) in the second 2d plane.

What could any complex phenomenon possibly be other than small “mundane” components combined together in a variety of ways and in immense quantities?

All such things are like this.

For me, this is fascinating, mind-boggling, non-sensical, and unsurprising, all at once.

But I wouldn’t call it inelegant.

> When I was in college they told us most of computing resources were used doing Linear Programming.

I seriously doubt that was ever true, except perhaps for a very brief time in the 1950s or 60s.

Linear programming is an incredibly niche application of computing used so infrequently that I've never seen it utilised anywhere despite being a consultant that has visited hundreds of varied customers including big business.

It's like Wolfram Mathematica. I learned to use it in University, I became proficient at it, and I've used it about once a decade "in industry" because most jobs are targeted at the median worker. The median worker is practically speaking innumerate, unable to read a graph, understand a curve fit, or if they do, their knowledge won't extend to confidence intervals or non-linear fits such as log-log graphs.

Teachers that are exposed to the same curriculum year after year, seeing the same topic over and over assume that industry must be the same as their lived experience. I've lost count of the number of papers I've seen about Voronoi diagrams or Delaunay triangulations, neither of which I've ever seen applied anywhere outside of a tertiary education setting. I mean, seriously, who uses this stuff!?

In the networking course in my computer science degree I had to use matrix exponentiation to calculate the maximum throughput of an arbitrary network topology. If I were to even suggest something like this at any customer, even those spending millions on their core network infrastructure, I would be either laughed at openly, or their staff would gape at me in wide-eyed horror and back away slowly.