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Questions tagged [multilinear-algebra]

For questions about the extension of linear algebra to multilinear transformations of vector spaces.

A multilinear function is a function from $V_1\times V_2\times\dots \times V_n\to V$ where the $V_i$ and $V$ are all vector spaces, and the function is a linear function into $V$ when restricted to each of the $V_i$. This includes the special case of bilinear functions.

For example, the ordinary dot product in $\Bbb R^n=V$ is a multilinear function from $V\times V\to \Bbb R$.

1327 questions
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Tensors: Acting on Vectors vs Multilinear Maps

I have the feeling like there are two very different definitions for what a tensor product is. I was reading Spivak and some other calculus-like texts, where the tensor product is defined as $(S \otimes T)(v_1,...v_n,v_{n+1},...,v_{n+m})=…
Squirtle
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Polarization formula.

Let $V$ be a $\mathbb{R}$-vector space. Let $\Phi:V^n\to\mathbb{R}$ a multilinear symmetric operator. Is it true and how do we show that for any $v_1,\ldots,v_n\in V$, we have: $$\Phi[v_1,\ldots,v_n]=\frac{1}{n!} \sum_{k=1}^n \sum_{1\leq…
Gilles Bonnet
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Subspace intersecting many other subspaces

V is a vector space of dimension 7. There are 5 subspaces of dimension four. I want to find a two dimensional subspace such that it intersects at least once with all the 5 subspaces. Edit: All the 5 given subspaces are chosen randomly (with a very…
karthik
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Determinant and alternating multilinear function

Let $V$ be a vector space of dimension $n$ with basis $\{v_1,\cdots,v_n\}$. Let $\phi$ be an n-alternating multilinear map and $A:V\rightarrow V$ is any map (matrix form) then we have to prove that $\phi(Av_1,\cdots,Av_n)=\det(A)…
user87543
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Questions about bilinear maps from a Gowers's article

The following is from Gowers's How to lose your fear of tensor products: Because subscripts and superscripts are a nuisance in html, I shall now change notation, and imagine that we know the values of $f(s,t), f(u,v), f(w,x)$ and $f(y,z)$. If $s,…
user9464
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Symmetric multilinear form from an homogenous form.

Let $V$ be a $n$-dimensional $\mathbb{R}$-vector space. Let $\phi:V\to\mathbb{R}$ a homogeneous form of degree $n$, i.e. $\phi(\lambda v)=\lambda^n \phi(v)$. If we define the symmetric multinear [!see edit!] operator $\Phi:V^n\to\mathbb{R}$…
Gilles Bonnet
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Poincaré duality and Hodge duality

The Poincaré duality is defined in Greub's Multilinear algebra (1967) in Chapter 6, §2 as a isomorphism between $\bigwedge V$ and $\bigwedge V^*$, where $V$ is a finite-dimensional vector space, $V^*$ is its dual, and $\bigwedge V$ is the exterior…
Andrey Sokolov
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How should I understand "every bilinear function $f$ on $V\times W$"?

In this article, there is a lemma as following: Let $U$ and $V$ be vector spaces, and let $b:U\times V\to X$ be a bilinear map from $U\times V$ to a vector space $X$. Suppose that for every bilinear map $f$ defined on $U\times V$ there is a unique…
user9464
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Do any non-combinatorial proofs of the elementary properties of wedge products exist?

The wege product, an operation defined between two alternating tensors, has a number of elementary properties such as associativity, distributivity, etc. There are many proofs of these properties e.g., see Analysis on Manifolds by Mukres or…
ItsNotObvious
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Multilinearity of the Polarization Formula

I'm going to asking a question about the polarization formula of an $n$-form. Given an $n$-homogeneous form $q:V\rightarrow \mathbb{R}$ on a vector space $V,$ i.e., $\forall v\in V, r\in \mathbb{R},$ we have $q(rv) = r^nq(v),$ the polarization…
User
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Exterior algebra of a subspace

Let $E$ and $E^\star$ be two vector spaces in duality according to a (possibly symmetric) non-degenerate bilinear form $\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb{R}$. Let $F$ be a subspace of $E$. I would like to show…
Benjamin
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Given math model to model relations using a tensor, how do we put restriciton on tensor so that it can capture symmetric relations

The goal is to model real life relations between stuff and people. Say we have sets $E$, $R$ and functions $h:E \to V$, $t :E \to W$ and $r:R \to U$, where $V,W,U$ are finite dimentional vector spaces with potentially distinct dimentions but all…
user 6663629
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Dimension of vector space of multilinear forms $V^n \rightarrow K$

A multilinear form is a mapping \begin{align} \Delta: V^n \rightarrow K \end{align} where $V$ is a finite-dimensional vector space over field $K$. It must meet the following requirements: First: \begin{align} &\Delta\left(a_1, \dots, a_{i-1},…
fpmoo
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A clarification regarding partial derivatives

Let us suppose the $i^{th}$ partial derivative of $f:\Bbb{R}^n\to \Bbb{R}$ exists at $P$; i.e. if $P=(x_1,x_2,\dots,x^n)$, $$\frac{f(x_1,x_2,\dots,x_n+\Delta x_n)-f(x_1,x_2,\dots,x_n)}{\Delta x_n}=f'_n (P)$$ My book says this implies that…
user92664
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Cannot decide a certain multilinear map is alternating

Suppose $\mathfrak{g}$ is a Lie algebra and $f$ is a multilinear map from $\mathfrak{g}^k$ to $\mathbb{R}$ and let $\operatorname{Alt}(f)$ be the alternization of $f$. Then consider the map $h: \mathfrak{g}\times\cdots\times \mathfrak{g} \to…
Uncool
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