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Suppose you have a matrix $A$. Is there a "standard"/mathematical elegant way to denote all members of the matrix as a set?

So suppose there is a matrix $A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] $ then I would like to define $set(A) = \{ a,b,c,d \}$

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    If you define your own notation and it becomes popular, everybody will speak about the dtech-notation for matrix-elements! ;p2011-03-29
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    We call them the "entries of $A$", and usually refer to them that way. So you would write $\{x\mid x\text{ is an entry of }A\}$.2011-03-29
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    $\cup_{ij}\{A_{ij}\}$, maybe.2011-03-29
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    There are some problems with this notation: in fact, something like $\{ a,b,c,d\}$ does not have "memory" of the place of each entry in the matrix (hence $A=\begin{pmatrix} a & b\\ c & d\end{pmatrix}$ and $B=\begin{pmatrix} a & c\\ d & b\end{pmatrix}$ have $set(A)=set(B)$ even if $A\neq B$), nor of the quantity of each entry in the matrix (therefore $I=\begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}$ and $M=\begin{pmatrix} 1 & 1\\ 1 & 0\end{pmatrix}$ have $set (I)= set (M)$ even if $I\neq M$)...2011-03-29
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    @Arturo Ok, $set(A)=\{x\;|\;x\,is\,an\,entry\,of\,A\}$ seem nice enough @Pacciu That is not a problem, that is just a consequence of transforming the matrix to a set. Clearly $set^{inv}$ can't exist and $set(A)$ might equal $set(B)$ even if $A \neq B$ but neither is required.2011-03-29
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    The notation reminds me of a paper I wrote involving Feynman diagrams. The diagrams are usually thought of as a set $\{a,b,c,d\}$, but there are advantages to assembling them into matrix form. This is because Feynman diagrams have to be compatible in order to multiply them and the matrix multiplication does the "sum over intermediate states" correctly. See http://arxiv.org/abs/1006.31142011-03-29
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    @Pacciu: It's a problem if you want your function to be one-to-one, but not a problem if you don't. For instance, when one proves that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(J)$ for some ideal $J$ of $R$, we do it by considering the ideal generated by the entries of the matrices, and we don't care *where* the entries are, just whether they are there *somewhere*.2011-03-29
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    @Arturo: Thanks. I suppose I saw those problems because I'm more familiar with matrix representations coming from Numerical Analysis (e.g., the ones for tridiagonal or sparse matrices), which usually have to "keep track" of each element.2011-03-29
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    @Pacciu: Of course; there are certainly situations where you want to keep track of them, possibly by mapping the matrices to vectors in some standard way; and it's certainly the case that sometimes people talk about "sets" and yet think that order is important, so it's good to bring it up, just in case.2011-03-29

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