Confusion with definition of transposed matrix using indexed familiy

Question

Background

So we have a matrix: $A = (a_{ij}) = \begin{pmatrix} a_{11} & \dots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \dots & a_{mn} \end{pmatrix} \in K^{m \times n}$

The transposed matrx is often simply defined by $A^\mathrm{T} = (a_{ji})$.

Why this confuses me:

i and j are just variables (you could also rename them to $l$ and $k$). The only information they contain, is that $1\le i\le m$ and $1\le j \le n$. What really matters is the use of the first or second placeholder. The term $(a_{ij})$ is essentially an index familiy, another writing/interpretation for a function $a: \{ 1, ...,m \} \times \{ 1, ... ,n \} ) \rightarrow K$ (where the first set is the index for the rows (for first placeholder) and the second set is the index for the columns (second placeholder)).

Having that in mind, given the above Matrix $A$, if i would write $(a_{ji})$, i would expect this: $(a_{ji}) = \begin{pmatrix} a_{11} & \dots & a_{1m} \\ \vdots & & \vdots \\ a_{n1} & \dots & a_{nm} \end{pmatrix}$. For $m \neq n$ this woudn't even be well defined, e.g. beeing $n<m$ because $a_{1m}$ wouldn't be defined then (because the second placeholder can only have an index $\in\{ 1,...,n \}$).

Question

I see why this doesn't make content-related sense, BUT understanding $(a_{ji})$ as $\begin{pmatrix} a_{11} & \dots & a_{m1} \\ \vdots & & \vdots \\ a_{1n} & \dots & a_{mn} \end{pmatrix} \in K^{n \times m}$ doesn't make formally sense for me, see above (the referring indexed family $(a)$ is already defined in the above sense).

Does someone understand my confusion and can help me fix it? I would really appreciate an explaination :).

(Example sources: german wiki for families and transposed matrix)

Answer 1

You could perhaps say that $A$ is the function $(i, j)\mapsto a_{i,j}$ and $A^T$ is the function $(i, j)\mapsto a_{j,i}$, with suitable domains.

Answer 2

Yes the notation $A^T$ = $(a_{ji})$ is inherently confusing, a better way to write it is $(A^T)_{ij}$ = $(A)_{ji}$.

Answer 3

If you know programming, think about "for" loops. To write down your matrix you do two "for" loops, first iterating over $i$ and then iterating over $j$, and going from top to bottom by rows. If you keep always in mind that the first index is for the row position and the second index is for the column position then you can convince yourself that the transpose coincides with the indexing $(a_{ji})_{ij}$.