Matrix Multiplication: Understanding its Origins and Order

Where does matrix multiplication come from?

What AB Represents

The product AB represents the transformation B followed by transformation A. That is a counterintuitive concept, which we are accustomed to writing from left to right. Unfortunately, matrices are defined by rules that require multiplication from right to left. If you think about it, when you write a function f times a different function g, what you’re really saying is apply g, then f. Multiplying matrices is done in reverse order from that used in standard mathematical notation.

If, for example, I write AB times X, where X is some vector that I want to transform, it’s the same as A times BX The associative property is a good property of well-behaved products. Therefore, matrix product is associative. This means that we can reorder the factors in a product ABX without changing the result. We can start with BX or we can start with AB. BX means applying transformation B to X, and multiplying by A means applying transformation A.

We first apply B, then we apply A or we can apply them both at once: AB. What AB represents: do transformation B, then transformation A. (AB X= A(BX)