In working on a research problem in order theory, I have encountered a symmetric rank-1 matrix that can be expressed as $$ A = 30 \begin{pmatrix} 1 \\\\ 1/5 \\\\ 1/2 \\\\ 1/6 \\\\ 1/15 \\\\ 1/30 \end{pmatrix} \begin{pmatrix} 1 \\\\ 1/5 \\\\ 1/2 \\\\ 1/6 \\\\ 1/15 \\\\ 1/30 \end{pmatrix}^\mbox{T} . $$ For reasons specific to my problem, I'm interested in the function $f$ that maps each $a_{ij}$ to the union of $\mbox{row}_i(A)$ and $\mbox{col}_j(A)$ (obviously taken as the sets of their respective entries).
Is this kind of function already a thing? If so, what's it called, and what are its known properties? I'd really like to be able to reason about the members of $f(a_{ij})$ for arbitrary $i$ and $j$. And just to be clear, the $i$'s and $j$'s are of no interest beyond their use as indices. What I'm interested in is $A$'s entries themselves, and how each one's value may relate to its image under $f$.
Of course, it wouldn't be a particularly onerous job to simply brute-force it and fish out the 36 sets by inspection---I know that for this particular matrix $A$, $|\{a_{ij}:1\leq i,j \leq 6\}| = 19$---but the extensional values of the $f(a_{ij})$'s aren't as illuminating as I'd like. And my gut tells me that there's something interesting lurking in this set-valued function. Ideally, I'd like to be able to construct $f(a_{ij})$ directly from $a_{ij}$ and without looking things up in $A$. I do have a way to acquire $i$ and $j$ from the value of $a_{ij}$, though it's more an algorithm than a formula.
