The "extended" dot product

Question

The standard dot product of two vectors $\vec{u},\vec{v} \in \mathbb{R}^n$ is given by:

$ \vec{u}\cdot \vec{v} = \sum_{i=1}^n u_i v_i$

Assuming now that I have $m$ vectors in $\mathbb{R}^n$, does the following product have a name, and does it have any interesting properties:

$P = \sum_{i=1}^n \prod_{j=1}^m v_{j,i}$

In other words, it's the extension to the dot product, which is defined as the sum of element-wise products of all vectors.

@WhatsUp $v_1 \cdot v_2$ is a number, how do you take the dot product of a number and a vector? — Najib Idrissi, May 30 '16 at 13:45
I don't think so, for the reason mentioned by @Najib Idrissi — Michael Stachowsky, May 30 '16 at 13:45
Sorry, I was wrong. This is an $m$-linear map from $(\mathbb{R}^n)^m$ to $\mathbb{R}$. — WhatsUp, May 30 '16 at 13:46
This is related. You can define whatever operation on vectors you want, but if it is not "geometric" (in the sense that it does not depend on the choice of coordinates) then it will probably be of marginal use. — Giuseppe Negro, May 30 '16 at 13:47
It seems that it is, in fact, a multilinear map. I'm interested in questions like: under what conditions is it zero for non-zero vectors $\vec{v}_j$? I'm not entirely sure if there is a nice answer, though — Michael Stachowsky, May 30 '16 at 13:49
@GiuseppeNegro I don't think the usual dot product is "geometric" in your sense. Neither do I think it is of marginal use. — WhatsUp, May 30 '16 at 13:57

Michael Stachowsky · Accepted Answer · 2016-05-30T14:26:25.947

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The answer is: it is a multilinear map. So yes it has a name, and yes it has interesting properties. The properties are the subject of study of multi-linear algebra, so are too vast to mention here.

edited May 30 '16 at 14:26

answered May 30 '16 at 14:23

Michael Stachowsky

2,943

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The question is "does the following product have a name, and does it have any interesting properties". So I think the answer should be one of the four: "yes, yes", "yes, no", "no, yes", "no, no". – WhatsUp May 30 '16 at 14:25
Edited and fixed :-) – Michael Stachowsky May 30 '16 at 14:26
Well ... I was just kidding ... but agree that it's too vast to talk about "interesting properties" here. – WhatsUp May 30 '16 at 14:27

The "extended" dot product

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