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My question: What is a notation for an empty 0x0 matrix (i.e. the matrix for the only linear map $f:\{0\}\to\{0\}$)? Is it written $()$? How can I distinguish the 0x0 matrix with for example the 0x3 matrix or the 3x0 matrix?

What I have already found out: Concerning the section “Empty matrices” of the Wikipedia article “Matrix (mathematics)” there “is no common notation for empty matrices”. But unfortunately I haven't found any notation so far...

Notes: I am looking for a notation which is used in a textbook. I am not interested in how empty matrices can be created in CAS like Matlab, Mathematica, etc.

Reason for this question: In our course we had the task to draw all graphs with three vertices and to state the incidence matrix for each drawn graph. Thus, for the empty graph I have to state a 0x3 matrix, but I didn't know the right notation for it...

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    $\begingroup$ You could call it $\mathrm{Id}_0$, to represent the $0 \times 0$ identity matrix. $\endgroup$ – Patrick Stevens May 10 '16 at 9:03
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    $\begingroup$ I like the notation $[]$, the "matrix with nothing in it" $\endgroup$ – Ben Grossmann May 10 '16 at 18:53
  • $\begingroup$ Just curious: Is there a mathematical reason you're looking to denote the $0 \times 0$ matrix? $\endgroup$ – Andrew D. Hwang May 13 '16 at 22:24
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    $\begingroup$ @AndrewD.Hwang: No, actually the reason for asking this question was how I can notate the empty 0x3 matrix. I will add my question with an explanation why I asked the question... $\endgroup$ – Stephan Kulla May 14 '16 at 7:08
  • $\begingroup$ For a graph on $n$ vertices, the incidence matrix is $n \times n$, and the $(i, j)$-entry is the number of edges from vertex $i$ to vertex $j$. For the empty (i.e., edgeless) graph on three vertices, an existing matrix suits your needs. :) $\endgroup$ – Andrew D. Hwang May 14 '16 at 11:06
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I don't think there is any universally recognizable notation for this. Whatever choice one might make, without explanation it could easily be misunderstood, is the point.

(For that matter, I can't help wondering how/why the notation would be needed in a context that wouldn't permit the simpler explanation "unique linear map from $k^0$ to $k^0$".)

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I think the most simple notation would be to write $$ 0_{M_{0,0}}$$ for a matrix of size $0\times 0$, and $$ 0_{M_{3,0}}$$ for a matrix of size $3\times 0$.

It's very similar when you want to write the zero of an unknown field $F$, when you write $0_F$.

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