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What's the most common way of writing the all-ones vector, that is, the vector, when projected onto each standard basis vector of a given vector space, having length one? The zero vector is frequently written $\vec{0}$, so I'm partial to writing the all-ones vector as $\vec{1}$, but I don't know how popular this is, and I don't know if a reader might confuse it with the identity matrix.

I'm writing for a graph theory audience, if that helps pick a notation.

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    $\begingroup$ "standard basis vector of a given vector space" To nitpick language: A "given" vector space need not have a standard basis. What you probably mean is that a basis for your vector space is fixed, or you are just considering the standard basis of $k^n$. The zero vector is different, because it is all zeros regardless of basis. However, I don't see anything wrong with your notation for all 1's so long as the basis is understood. If it doesn't seem too cumbersome and you want to be careful, you could write $\sum_{k=1}^n e_k$ if your basis is $e_1,\ldots,e_n$. $\endgroup$
    – Jonas Meyer
    Dec 27, 2009 at 21:29
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    $\begingroup$ This question should be closed for being too localized. $\endgroup$
    – Harry Gindi
    Dec 27, 2009 at 21:33
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    $\begingroup$ I think the question is fine. $\endgroup$
    – Greg Kuperberg
    Dec 27, 2009 at 21:35
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    $\begingroup$ The "all ones vector" has no meaning without choosing a basis, of course, because there's no canonical choice of a "1" vector. However, if you view your vector space as a direct product of fields, you can use the notation 1, since this element is the unit of the ring. It is terribly misleading, however, to use the notation 1 if you don't care about the ring structure. $\endgroup$
    – Harry Gindi
    Dec 27, 2009 at 21:46
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    $\begingroup$ There sure are a lot of (pedantic, I would say) comments regarding the choice of basis. Since this seems to be in a graph theory context, perhaps the vector is being used to denote some sort of incidence information. In that case, we do have a preferred basis: that corresponding to the vertices of the graph, perhaps. $\endgroup$
    – Kevin H. Lin
    Dec 28, 2009 at 10:41

7 Answers 7

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I have used the notation $\vec{1}$ in a paper. I think that it's a good choice if you help the reader by defining it. I did a Google Scholar such of "vector of all ones", and I found a lot of so-so notation such as $e$, $u$, $\mathbf{e}$, $\mathbf{1}$, and even just plain $1$. I don't think that the literature is loyal to any particular choice. Confusing $\vec{1}$ with a matrix would be a little strange, because a matrix is suggested by a two-headed arrow, or $\stackrel{\leftrightarrow}{1}$.

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  • $\begingroup$ e and bolded e seem acceptable to me. They are good for the following reason: this notion only makes sense in light of a choice of basis and norm, and the notations using the letter e make it clear that this is only with respect to the basis $\{ e_i \}_{i\in I}$ $\endgroup$
    – Harry Gindi
    Dec 27, 2009 at 22:05
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    $\begingroup$ My feeling is that each of the other notations could mean any number of different things. By contrast, it's hard to avoid the intended meaning of $\vec{1}$ when, in context, there is a distinguished basis. $\endgroup$
    – Greg Kuperberg
    Dec 27, 2009 at 22:26
  • $\begingroup$ As much as I love the bold 1 notation, I think I will go with the bold e, since that appears popular in the papers I am citing. $\endgroup$
    – Bkkbrad
    Dec 27, 2009 at 23:23
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I like \mathbb'ed ones for this. You can use the mathbbol package by simply saying \mathbb{1}.

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    $\begingroup$ I use \mathbf{1} to make a bold $\mathbf{1}$ instead of \mathbb{1} as $\mathbb{1}$. $\endgroup$
    – Nick Dong
    Jan 14, 2019 at 12:34
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I use \mathbf{1} in papers (and in books) In combinatorics it is also common to use $j$, and to use $J$ for the all-ones matrix. Using $j$ for the all-ones vector has obvious problems since it occurs so often as an index. No solution is perfect, but I find I have fewer problems with \mathbf{1}.

I agree you should define it.

Generally I avoid using decorations (tildes, arrows,...) to represent vectors - they look really ugly on the page.

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  • $\begingroup$ You should use arrows for vectors for the following reason: Unless your handwriting is impeccable (unable to be pecced, presumably), it is sometimes very difficult for students taking notes to tell whether or not the "v" that you've written is capital or lowercase, and hence a vector or a vector space. $\endgroup$
    – Harry Gindi
    Dec 27, 2009 at 21:51
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    $\begingroup$ I find that a tilde works fine on a blackboard. $\endgroup$
    – Chris Godsil
    Dec 28, 2009 at 3:11
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    $\begingroup$ @HarryGindi, although (especially 5 years later) this is hardly the point, I have to mention that it literally means "unable to sin" ('peccare' being 'to sin' in Latin). $\endgroup$
    – LSpice
    May 18, 2015 at 18:49
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Let $I \subset \{ 1,2,3,\ldots, n \} $. Let $e_I = \sum_{i\in I} e_i.$ Let $[n]=\{ 1,2,3, \ldots, n \} $. Then $\vec{1}=e_{[n]}$. Also $e_{\{i\}} = e_i$. This is not satisfactory to your context, but may have the advantage of alternative usages in subsequent contexts.

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Once I had the same problem, I used notation similar to yours: $\mathbf{0}$ for zero-vector and $\mathbf{1}$ for "all-ones vector".

  • It is NOT common, so you have to define it

  • I would not do it unless you have many formulas with it --- if you use it just few times denote it by some letter...

Postscritp. Often $x*x*\dots*x$ is denoted by $x^{*n}$, so you may use $1^{,n}$ for $1,1,\dots,1$.

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Clearly there's no consensus on this issue. Personally, I dislike bold-face anything in papers as it's often hard for the reader to tell whether it's bold-face or not (not everyone has a decent printer + good eyesight). I would use $\vec{1}$ myself, but it doesn't matter so much, as long as its defined appropriately.

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I'd say, denote it any way, but please make clear in the introduction that it depends on the basis!

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