[whywhywouldyou]

Came here to say the same thing. This adds absolutely nothing to any reasonable understanding of matrix multiplication. This is the most complex way I could imagine trying to explain what matrix multiplication "is".

If it's useful to someone working in a very complex environment where these visualizations are necessary to help tease out some subtle understanding, then that's great.

But really, this part is all you need to know about the article:

> This is the _intuitive_ meaning of matrix multiplication:

> - project two orthogonal matrices into the interior of a cube

> - multiply the pair of values at each intersection, forming a grid of products

> - sum along the third orthogonal dimension to produce a result matrix.

This 1. Isn't intuitive, and 2. Isn't the "meaning".

[sdwr]

You kidding me? This is the only reasonable explanation of matrix multiplication. I remember learning it in school, and memorizing how the input dimensions corresponded to output dimensions, without understanding why.

This gets to the why perfectly. We all understand how to navigate a 3D space intuitively. If the math doesn't tie into that, it may as well be wizard nonsense.

[ndriscoll]

The only reasonable explanation of matrix multiplication is that

1. For a linear function f, its matrix A for some basis {b_i} is the list of outputs f(b_i). i.e. each column is the image of a basis vector. For an arbitrary vector x, the matrix-vector product Ax = f(x).

2. For two linear functions f,g with appropriate domains/codomains and matrices A,B, the result of "multiplication" BA is the matrix for the composed (also linear) function x -> g(f(x)). For an arbitrary vector x, the product (BA)x = B(Ax) = g(f(x)).

This tells you what a matrix even is and why you multiply rows and columns in the way you do (as opposed to e.g. pointwise). This also tells you why the dimensions are what they are: the codomain has some dimension (the height of the columns) and the domain has some dimension (how many columns are there). For multiplication, you need the codomain of f to match the domain of g for composition to make sense, so obviously dimensions must line up.

[Rayhem]

> This gets to the why perfectly.

Respectfully disagree. A matrix has basically nothing to do with "living on the surface of a cuboid". It's like saying FOIL is the "why" of binomial multiplication -- the "why" is the distributive and associative properties of the things involved, FOIL is just a useful mnemonic that falls out.