Matrix Multiplication in APL

I’ve been working on linear algebra in APL and I decided to write an article on one of the fundamental operations of linear algebra (and neural networks), matrix multiplication! In this post I walk through how to multiply a 2x2 array by a 2x1 array.

And there’s no matrix multiplication glyph :scream:!

This post walks through:

  • What is matrix multiplication and manual calculation in this simple example
  • How to do matrix multiplication in APL step by step and how and why it works (including explaining the dot glyph)
  • Shows a first (relatively) simple example of how glyphs will combine in a really powerful and flexible ways. Matrix multiplication for example is +.x, rather than having a glyph specific to matrix multiplication.

Hope you enjoy!

Link to post


Great stuff. I’m a big fan of inner product in APL. So cool the way they generalised the idea beyond standard math notation

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Hey all - I got a question about how this works with a matrix times a matrix instead of a matrix times a vector, so I updated the post to include an example showing that it works the same way with 2 matrices!

@Ezno I believe that, because inner product in APL is an operator, not a function, the arguments to it are ⍺⍺ and ⍵⍵, not and .

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I did not know that. I’ve updated the post to use that convention. Thanks!

Really cool Isaac! :smile: (btw the bullet points in your introduction don’t seem to be parsing properly- I’m viewing on mobile)

I was wondering if someone could help me in understanding the following example:

    (2 2 ⍴ ⍳4) +.× (2 ⍴ 2)
6 14


    2 2 ⍴ ⍳4 +.× 2 ⍴ 2
1 2
3 4

I understand the first example, but I am unsure what is going on in the second. I assume it is something with precedence but I am not sure.

APL does right to left precedence. So 2 2 ⍴ ⍳4 +.× 2 ⍴ 2 would be equivalent to (2 2 ⍴ (⍳4 +.× (2 ⍴ 2)))