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I understand $y_{ij}$ can be used to represent a cell in a matrix (i.e., the value in row $i$ and column $j$), particularly where the length of $j$ is equal for all $i$. I know it can also be used to index a vector $y_{i.}$ where each element is a vector indexed by $j$, and that this can permit different lengths of $j$ for each $i$.

  • What is the general name of this mathematical structure?
  • Should I think of it as a matrix, a vector of vectors, a long data frame (with one column for each index and one column for the value) or something else?
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    Thanks everyone. Your responses are helpful. My question probably reflects the fact that I'm a relative novice at mathematics. When reading journal articles in statistics, I get taken out of my comfort $z$one. I've seen a wide range of applications of index notation, and I was loo$k$ing for some form of general guidance about the essence of this notation, and perhaps, how to thin$k$ about it.2011-12-09

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If you consider your matrix as a linear operator $T:V\to W$, these entries are simply elements of the scalar field your vector space is over. Without restrictions on $T\in M_{n\times m}(\mathbb{F})$, the set of all elements with this property is simply given by $\{y_{ij}\}= \mathbb{F}$.

If the $j=j_0$ for all $i$ in a particular column, you can refer to these entries as the $j_0$th column vector.

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    Ah, I think I see what you mean. Are you saying that you want to categorise all of the $y_{ij}$'s for a particular $j$. I recently just made an edit that should help with that. If not, just give us a yell, and I can try to throw some more stuff in.2011-12-09