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I've seen J referenced recently in some discussion about algebraic combinatorics, and it took me a while to figure out it was the matrix of all ones. It came up without definition, and I spent too long trying to google "what is a J matrix".

I found this, but I can find no reference to the convention of using J to represent the matrix. Maybe it's arbitrary?

https://en.wikipedia.org/wiki/Matrix_of_ones

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    $\begingroup$ One guess is that the identity matrix already has a well-established claim on the more natural-seeming "I", so "J" is the next alternative. $\endgroup$
    – Noam D. Elkies
    Jan 24, 2019 at 3:11
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    $\begingroup$ I see $E$ in some 1950s sources. $\endgroup$
    – Brendan McKay
    Jan 24, 2019 at 8:24
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    $\begingroup$ At least it is not an 'O' for "all Ones". $\endgroup$
    – Per Alexandersson
    Jan 24, 2019 at 11:03
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    $\begingroup$ Perhaps it's from Jednostka, the Polish word for unit. $\endgroup$
    – Gerry Myerson
    Jan 24, 2019 at 11:38
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    $\begingroup$ See the related question (which has a bunch of answers, then ended up being closed) mathoverflow.net/questions/9898/… $\endgroup$
    – Joe Silverman
    Jan 24, 2019 at 12:48

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You have a matrix only filled with 1s, so it would be neat to have a notation resembling this. You could use

  • $1$, $\mathbf 1$ or $\mathbb 1$ but these are usually a danger for confusion with the scalar and the latter one is often used to represent the all-1s-vector already.
  • a letter similar to 1, e.g. $I$, $\mathbf I$ or $\mathbb I$, but these are usually used for the identity matrix already.

So you go with the next best letter that resembles a "1" $-$ namely $J$. However, I also have seen $E$. I would say that if you have some experience in the topic, you can spot this from the context. But a well-written text would introduce this notation nevertheless.

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    $\begingroup$ Unfortunately $E$ is also commonly used for a matrix with only one non-zero entry (often subscripted, like $E_{i j}$). $\endgroup$
    – LSpice
    Jan 24, 2019 at 14:57
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    $\begingroup$ If we are going to be unfortunate, then J is also used in almost-complex geometry/analysis to stand for a completely different matrix that squares to -1 (called the "symplectic matrix" or something like that). $\endgroup$
    – Linas
    Jan 24, 2019 at 18:14

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