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For a matrix $A$ in, say for instance, $\mathbb{R}^{m \times n}$, there are at least two different conventions to denote its entry at position $(j,k)$:

  1. Denote the entry as $a_{jk}$.

  2. Denote the entry as $A_{jk}$ .

In my experience (at several universities in Germany) the first convention is much more common in teaching; but this might be different in other places.

When teaching a linear algebra course last year I chose to use notation 2, though. The reaction of some colleagues (who were, let's say, surprised) and the prevalence of the first notation in many books I looked up, made me wonder whether I might be overlooking some important advantages of notation 1 or disadvantages of notation 2. So for my decision about the notation in future courses, I'm interested in the following:

Question. What are advantages and disadvantages of notations 1 and 2, respectively, in math teaching?

Remarks:

  • I am not asking for your opinion on which notation is "better". Instead, I am asking for a list of advantages and disadvantages, which can be used by course instructors to make an informed decision for their individual courses.

  • The question is mainly about the advantages and disadvantages in teaching (thus I am asking this on Math Educators Stackechange). The same question might be relevant for research papers, but this is not the focus of this question.

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  • $\begingroup$ What if I want the matrix to be $a$ ? Then what are its components if we insist on lower case for components. I think the convention is largely a typesetting convenience for books. I much prefer to use the capital letter because it aligns with the general conventions for assigning indices to tensors. $\endgroup$ Sep 27 at 15:31

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Sometimes a matrix name is suggestive: for example Jacobian or Ricci. We might use $\text{Jac}$ or $J$, or $\text{Ric}$ or $R$. In these situations it would be awkward to switch to lower case to reference an entry; it would be linguistically awkward to write $r_{i,j}$ instead of $R_{i,j}$ if $R$ is meant to refer to Ricci curvature.

Another example is the use of something like $\Omega$ to name a matrix, e.g., a curvature two-form. Strict adherence to the convention of switching to lowercase font for entries of $\Omega$ would dictate we use $\omega_{i,j}$ for an entry of $\Omega$. For those less comfortable with Greek characters, this would be confusing. I have never seen the Greek letter $\Xi$ used to name a matrix, but its lowercase variant is $\xi$ and then using $\xi_{i,j}$ for an entry of $\Xi$ seems ill-advised.

If your students are going to use an environment such as MATLAB or Python, then adherence to a notational convention of switching to lowercase font to designate entries is not natural. In programming, if we have a matrix $A$ then an entry is specified typically by function notation $A(i,j)$.

These examples not withstanding, using $a_{i,j}$ to denote an entry of $A$ is not incorrect. Mathematical notation is often ambiguous, and learning to adjust to notational ambiguities is something that needs to be learned. But your second convention seems pedagogically more sound.

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$A_{jk}$ is sometimes used to mean the matrix $A$ with row $j$ and column $k$ deleted.

[For example, see David Lay, Linear Algebra and its Applications, 4th edition, page 165.]

To avoid confusion with that usage, I would never use the capital letter to mean an entry.

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    $\begingroup$ Yes, but the capital (or, rather, the same) letter notation can be beneficial too: it is very convenient just to write $(A+B)_{ij}=A_{ij}+B_{ij}$ and $(AB)_{ij}=\sum_k A_{ik}B_{kj}$ when introducing the sum and product rules. What Lay denotes by $A_{ij}$ has then to be denoted by $A^{(ij)}$ or something like that, of course, to avoid the confusion you are talking about. $\endgroup$
    – fedja
    Sep 24 at 0:12
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    $\begingroup$ You are both right, but Sue's answer is definitely a possible reason not to use that notation. Hefferon's and Nicholson's (free) texts use the same convention, so in North America this might be just as much the case as the OP says it is in Germany. $\endgroup$
    – kcrisman
    Sep 24 at 2:38
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    $\begingroup$ There is also a convention to use $A^{jk}$ (rather than $A_{jk}$) to denote the matrix obtained from $A$ by deleting $j$th row and $k$th column... :) $\endgroup$ Sep 25 at 18:46
  • $\begingroup$ @paulgarrett And in TeX, if read in Latin, in which ^ = caret = “is missing”, A^{jk} reads “A is missing jk.” :) $\endgroup$
    – Raciquel
    Oct 1 at 22:55
  • $\begingroup$ @Raciquel, :) ... $\endgroup$ Oct 1 at 23:03
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One area where capitals may be useful is statistics, where you'll encounter matrices whose entries are random variables and denoted $X_{ij}$. Generally I prefer lowercase but that's probably out of habit; I don't see what's so bad about calling entries $A_{ij}$ other than that it's less common.

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The convention you're describing isn't used in physics. The entries of matrix $A$ are $A_{jk}$. Part of the reason for this is probably that physicists tend to assign specific physical interpretations to the various letters of the alphabet. For instance, $I$ would be a moment of inertia, which can be represented by a matrix, while $i$ would be either an electrical current or $\sqrt{-1}$.

Relativists also often use abstract index notation, in which $A_{mn}$ would actually be the entire matrix, without reference to any basis, while $A_{\mu\nu}$ is an element in a specific basis. (Latin letters are abstract indices, Greek letters are concrete.) To understand the logic of this in a more familiar context, consider a notation like $x^2-1$ to mean a function. You can think of $x$ as being a notation for the identity function. Abstract index notation is similar. Abstract indices don't have values like 3, just as the identity-function $x$ doesn't have a value like 3.

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If you are teaching a lot of students whose first language doesn't use the Roman alphabet, the bijection between upper and lower case letters may be something they have to pause and think about. I've run into this with Chinese students.

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