Kind of like a unit vector, but it's a matrix.

For example, if the notation is $A_{ij}$, then this matrix has all zero elements, except $a_{ij} = 1$.

Is there a common notation/term for this kind of matrix?

The goal is to write something like: for a matrix $B$, then $B + A_{12}$ gives the same matrix but one element at $(1,2)$ is incremented.

Is there a better notation for this goal?


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    $\begingroup$ I have seen $e_{12}$ $\endgroup$ – J. W. Tanner May 22 at 23:49
  • $\begingroup$ Such matrices form a basis of matrices as a vector space $\endgroup$ – J. W. Tanner May 22 at 23:51
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    $\begingroup$ The Wikipedia article Single-entry matrix uses $J^{ij}$ notation but I have not seen it before. More common is Elementary matrix $L_{ij}(1) = I_n +J^{ij}.$ Another choice is $e_i\otimes e_j^* = J^{ij}$ using tensor product. $\endgroup$ – Somos May 23 at 0:13
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    $\begingroup$ I have seen and used $E_{12}$. $\endgroup$ – lhf May 23 at 1:02
  • $\begingroup$ @Somos Ah neat, didn't find this article! Although the article itself admits the name (and possibly notation) is "not common" haha $\endgroup$ – smörkex May 23 at 1:25

You could use any letter you want to use for a single-entry matrix, as long as you define it, but $e$ is a nice choice (e.g., $e_{12}$), since $e$ is often used for basis vectors, and the single-entry matrices form a basis of matrices as a vector space.

  • $\begingroup$ This is what I had thought too, although matrices are commonly capitalized. However since basis vectors don't appear in the text this is a good choice. Thanks! $\endgroup$ – smörkex May 23 at 1:24
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    $\begingroup$ This notation is reasonably common in the literature, too. See, e.g., p. 2 of Humphrey's Introduction to Lie Algebras and Representation Theory. $\endgroup$ – Travis May 23 at 1:31

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