# Is there a common notation for a matrix with one entry equal to one and zero otherwise?

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!

• I have seen $e_{12}$ – J. W. Tanner May 22 at 23:49
• Such matrices form a basis of matrices as a vector space – J. W. Tanner May 22 at 23:51
• 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. – Somos May 23 at 0:13
• I have seen and used $E_{12}$. – lhf May 23 at 1:02
• @Somos Ah neat, didn't find this article! Although the article itself admits the name (and possibly notation) is "not common" haha – 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.