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?
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$\begingroup$ I see no reason/benefits to use the longer version. I would guess writing $C$ or whatever it tradition. $\endgroup$– Michael BächtoldJan 7, 2020 at 21:23
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1$\begingroup$ 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. $\endgroup$– user507Jan 7, 2020 at 22:51
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2$\begingroup$ $C_{ij}$ 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 $AB$. $\endgroup$– Xander Henderson ♦Jan 8, 2020 at 1:28
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1$\begingroup$ @XanderHenderson but it requires introducing an arbitrary new letter and writing an additional equation, so I don't see it as more concise. $\endgroup$– Michael BächtoldJan 8, 2020 at 7:37
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2$\begingroup$ 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. $\endgroup$– Xander Henderson ♦Jan 8, 2020 at 13:26
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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.
Many students in an introductory linear algebra class, including many of those who pass the class, never fully assimilate the definition of matrix product, nor are capable of recapitulating it completely correctly at the semester's end (those who pass can all realize it operationally, but that is a different thing). One sees this starkly if one asks them to write a program to calculate the matrix product. Writing loops that implement the sum is a problem that separates those who will pass an introductory programming class from those who won't. Many students who do fine in linear algebra struggle with writing loops to implement the matrix product.
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2$\begingroup$ "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 $C$? $\endgroup$ Jan 8, 2020 at 9:25
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1$\begingroup$ @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. $\endgroup$– Dan FoxJan 8, 2020 at 10:25
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1$\begingroup$ 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. $\endgroup$ Jan 8, 2020 at 11:41
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$\begingroup$ @MichaelBächtold: There is a subtle difference; there is a difference between referring to the components of an undefined matrix and referring to the components of the undefined product of existing matrices. Exactly, what is undefined has a different character. However as I wrote above, I'm not trained in logic and don't know how to frame/state the difference in a precise manner, while on the other hand, my point is really that I don't think it is worth fussing about, particulary when explaining such things to students. $\endgroup$– Dan FoxJan 9, 2020 at 11:24