# Notation in the definition of matrix multiplication

When matrix multiplication is introduced, it is usually introduced with an additional variable: Given two multiplicable matrices $$A$$, $$B$$, one defines the product $$C=AB$$ to be the matrix given by some formula for the coordinates of $$C$$ ($$c_{ij} = \dots$$). Instead one could also define directly $$(AB)_{ij} = \dots$$ where $$(AB)_{ij}$$ are the entries of the product matrix. Every book I checked uses the longer notation, so is there a reason to use the long version? Which version is more beneficial for students?

• I see no reason/benefits to use the longer version. I would guess writing $C$ or whatever it tradition. – Michael Bächtold Jan 7 '20 at 21:23
• Your version might be harder to understand for a beginner who doesn't know what's being notated. The same thing comes up in discussing the cross product. – Ben Crowell Jan 7 '20 at 22:51
• $C_{ij}$ is more concise, and requires no parentheses for disambiguation. My guess would be that this is enough to justify the use of a single letter $C$ rather than the longer and potentially ambiguous $AB$. – Xander Henderson Jan 8 '20 at 1:28
• I always use $(AB)_{ij}$. Then I take off lots of points when students fail to distinguish between $AB_{ij}$ and $(AB)_{ij}$. Only one of these expressions lacks ambiguity. – James S. Cook Jan 8 '20 at 2:24
• You only have to write "Define $C = AB$" once. However, if you don't do this, you have to write $AB$ many times over the course of a discussion. This is demonstrably more concise. I'm not claiming it is better (I have no dog in that fight). I am simply observing that, in the long run, it requires less ink. – Xander Henderson Jan 8 '20 at 13:26

In class I define the matrix product by $$(AB)_{ij} = \sum_{k = 1}^{q}A_{ik}B_{kj}$$ where $$A$$ is $$p \times q$$ and $$B$$ is $$q \times r$$. In my experience what causes the most confusion is not the subscripts indicating the component, but the meaning of the summation notation. Students do not identify the written formula with $$(AB)_{ij} = A_{i1}B_{1j} +A_{i2}B_{2j} + \dots + A_{iq}B_{qj}$$. Also they are confused by the difference between dummy indices such as $$i$$, $$j$$, $$k$$, and numbers such as $$1$$, $$2$$, etc. Worse still, $$q$$ is used to represent a definite number, and not a dummy index (its value is not fungible).
The reason for defining $$C_{ij}$$ and then defining $$AB = C$$ (in this order, not the other) is a sort of logical fussiness that is misguided at the level where these things are taught. I am not trained in logic so I don't know how to state this carefully, but the issue is that a priori the product $$AB$$ is undefined, so referring to its components is somehow unsanitary. The point is that one is not supposed to refer to the components of something not yet specified. The solution is to write $$C$$ for the putative undefined product, specify its components and a posteriori declare that the matrix so defined is the product of $$A$$ and $$B$$. I suppose this is formally more correct (in some sense I am not able to verbalize), but for students who are struggling to understand the operational meaning of the sum involved in the definition, it is at best a distracting subtlety, and a worst a source of additional confusion.
• "The point is that one is not supposed to refer to the components of something not yet specified." Wouldn't the same then apply to $C$? – Michael Bächtold Jan 8 '20 at 9:25
• @MichaelBächtold: $C$ is defined by $C_{ij} = \sum_{k = 1}^{q}A_{ik}B_{kj}$. This clearly defines a matrix. The product of $A$ and $B$ is then defined to be the matrix $C$. Personally, I find this to be excessively fussy and am willing to forego the intermediary $C$, but I guess something like this is what the books which the OP mentions do. They find problematic writing $(AB)_{ij}$ when the matrix $AB$ is not yet defined. – Dan Fox Jan 8 '20 at 10:25
• I still don't see why you could not also claim that "It's problematic writing $C_{ij}$ when the matrix $C$ is not yet defined" or "$AB$ is defined by $(AB)_{ij}=\sum_{k=1}^q A_{ik}B_{kj}$". Just swapping $C$ and $AB$ in what you wrote. – Michael Bächtold Jan 8 '20 at 11:41