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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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Just call it either an "entry" (scalar) or a "block" (general). It is useful to thing of a matrix either in terms of its entries or in terms of its blocks, depending on the application. –  J. M. Dec 9 '11 at 1:09
    
It usually goes under the name of index notation –  user13838 Dec 9 '11 at 1:13
    
I'm not sure why a name is important. Unless there is something particularly interesting about the entries, they're really just elements of the scalar field. –  Henry Shearman Dec 9 '11 at 1:17
    
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 zone. I've seen a wide range of applications of index notation, and I was looking for some form of general guidance about the essence of this notation, and perhaps, how to think about it. –  mycat Dec 9 '11 at 5:07

1 Answer 1

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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Excuse my ignorance, but is this also true if $y_{ij}$ has a different number of $j$ for each $i$? For example if it were representing measures of individuals indexed by $i$ on a variable number of time points $j$? –  mycat Dec 9 '11 at 1:22
    
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. –  Henry Shearman Dec 9 '11 at 1:26

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