1
$\begingroup$

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?

Thanks!

$\endgroup$
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.