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I can think of multiple ways of writing the sum of a vector $\mathbf{v}$'s elements, but is there a standard operator for this?

Using "programming" notation it is typically sum($v$), but this seems informal. Perhaps just $\mathbf{1}^T \mathbf{v}$ or $\mathbf{1} \cdot \mathbf{v}$?

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    $\begingroup$ I don't know of a standard way to do this, and personally I would just go with sum(v). $\endgroup$
    – Ruben
    May 14, 2014 at 0:06

3 Answers 3

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$\sum_{i=1}^n v_i$ is the most widely accepted and understood notation.

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  • $\begingroup$ Would you use it as a function parameter? $\endgroup$
    – kon psych
    Oct 28, 2016 at 3:57
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    $\begingroup$ @konpsych You mean $f(\sum_{i=1}^n v_i)$? Sure, that seems correct and readable (and even more with \displaystyle and properly-sized parentheses). $\endgroup$ Oct 28, 2016 at 6:25
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If you use the euclidean 1-norm and you want the sum of the absolute values of the entries of $\vec{v}$ (or $v_i\geq 0$ ) you can take $\| \vec{v}\|_1$. Otherwise you can use $sum\{\vec{v} \}$ or $trace( diag\{ \vec{v}\})$. But I guess $sum\{\vec{v}\}$ would be the more "standard" notation.

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For the Euclidean space $R^n$, where the inner product is given by the dot product: $$\langle(x_1, x_2, ..., x_n), (y_1, y_2, ..., y_n)\rangle = x_1 y_1 + x_2 y_2 + ... + x_n y_n$$ See Inner Product - Wolfram MathWorld, http://mathworld.wolfram.com/InnerProduct.html

So we can use $\vec{1}$ to form a inner product with vector $\vec{v}$: $$\langle\vec{1}, \vec{v}\rangle = v_1 + v_2 + ... + v_n$$ or $$\langle\vec{v}, \vec{1}\rangle = v_1 + v_2 + ... + v_n$$

Pay attention to the dimension of $\vec{1}$, which be the same as that of $\vec{v}$.

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  • $\begingroup$ Hi @EdwardXu, welcome to Math SE! This doesn't seem to answer the question: it appears that the OP understands how $\mathbf{1} \cdot \mathbf{v} = \sum v_i$, and they actually want to know what is/are the standard notation/s for $\sum v_i$. If you feel that $\mathbf{1} \cdot \mathbf{v}$ is the standard notation then do mention this in your answer along with some source to back it up. $\endgroup$
    – user279515
    Feb 9, 2019 at 10:49
  • $\begingroup$ You are right. I realized that I didn't answer the question. But the inner product just come into my that it can be used to express the sum of all the product of the corresponding elements in two vectors. If one of the vector is identity vector, then the result is the sum of all the elements in the other one. $\endgroup$
    – Edward
    Mar 14, 2019 at 18:14

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