Questions tagged [multilinear-algebra]

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

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14 views

A perfect pairing with respect to the symmetric product

Let $V$ be a finite dimensional vector space endowed with a non-degenerated symmetric quadratic form $q$. Let $n\geq 1$ be a positive integer and $I_n$ the ideal of $\mathrm{Sym}^*V$ generated by ...
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64 views

Coderivative of symmetric $2$-tensor

This is taken from a paper by M. Berger: Here $(X,g)$ is a Riemannian manifold and $h$ is a symmetric $2$-tensor on $X$. Could you help me to understand this definition, please? What does $\nabla^k$ ...
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Natural Lie algebra structure on $\bigwedge\nolimits^2 \mathbb{R}^n$

As vector spaces, the Lie algebra $\mathfrak{so}(n)$ is isomorphic $\bigwedge\nolimits^2 \mathbb{R}^n$ with the isomorphism given on simple bivectors $$\Phi: \bigwedge\nolimits^2 \mathbb{R}^n \to \...
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Berger notation for a triple product of tensors

The following definition is given in a paper by M. Berger: Here, $(M,g)$ is a Riemannian manifold and $S^2(M)$ is the space of symmetric $2$-tensors on $M$. Is there a coordinate-free equivalent for ...
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On the solution space of linear matrix equations.

Consider a linear matrix equation $$ Y = \sum_{i} A_i X B_i^T = \sum_i (B_i\otimes A_i) \cdot X = T\cdot X $$ When does the solution space admit a basis consisting only of rank-1 matrices?
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Why are Tensors (Vectors of the form a⊗b…⊗z) multilinear maps?

In our linear algebra course we defined the Tensor Product between two vector spaces as an Operation so that this Diagram commutes: Here $\varphi$ and $\iota$ are multilinear maps and $\psi$ is a ...
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61 views

An svd-like tensor decomposition splitting a tensor into two lower-dimensional tensors and a singular vector

I have also posted this question on mathoverflow, but it seems there are a lot of questions related to SVDs here and a tag "tensor-decomposition", so I will give it a shot. I am looking for ...
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60 views

How to find Nullspace of Tensor / Kernel of a multilinear map?

[ Disclaimer: I am only starting to get my head around concepts of tensor calculus, so I apologize in advance for the lack of clarity or asking something trivial. I came across it trying to apply it ...
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13 views

Uniqueness of second -and fourth-order moment tensors of vectors

Let $\{v_1, v_2, v_3\}$ and $\{w_1,w_2,w_3\}$ be two sets of vectors in $\mathbb{R}^2$; we can assume the $v$s are pairwise linearly independent, and likewise for the $w$s. I'd like to show that if $$\...
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118 views

Stalks of exterior power

Assume we have a ringed space $(X,\mathcal{O}_X)$ and an $\mathcal{O}_X$-module $\mathscr{F}$. Then I want to see that for all $x\in X$ we have an isomorphism $$(\bigwedge_{\mathcal{O}_X}^r\mathscr{F})...
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25 views

non-vanishing of a symmetric tensor

Let $V$ be a finite dimensional vector space over $\mathbb{C}$. For any set of nonzero vectors $(v_1, v_2, \ldots, v_N )$, we can construct a symmetric tensor $$ W = \sum_{\sigma\in S_N }v_{\sigma(1)} ...
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64 views

What is the generalization of Pascal's formula for multi indices?

I am taking a course in PDE and trying to get use to these notations. If I take 3 vectors with nonnegative integers components (n components) that are denoted by $\alpha$,$\omega$,$\gamma$. Is it ...
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1answer
48 views

On (sums of) multivectors and linear independence

I have a doubt about multivectors and linear independence. Let $w_{1}$, $w_{2}$, $v_{1}$, $v_{2}$, $v_{3}$ be elements of some $n$-dimensional vector space, and assume that $v_{1} \wedge v_{2} \wedge ...
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1answer
31 views

What is the inclusion of this symmetrized space in $V \otimes \Lambda^2 V$?

This question concerns Exercise 2.8.1(1) in J.M. Landsberg's book Tensors: Geometry and Applications. Let $V$ be a $\mathbb{C}$-vector space, and consider the vector space which is defined as the ...
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126 views

Prove that $\varphi (v,\cdots, v)=\sum _{|\alpha |=k}\frac{k!}{\alpha !}v^\alpha \varphi^\alpha $

Firstly consider the following notations: Given any $\alpha =(\alpha _1,\cdots, \alpha _m)\in \mathbb{N}_0^m$ we define: $\color{red}{|\alpha|}:=\sum_{i=1}^m\alpha _i$ $\color{red}{\alpha !}:=\alpha ...
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1answer
61 views

Is the alternatization of a m-form on a m-dimensional vector space non-zero if there exists a basis on which the m-form is non-zero?

Let $V$ be a $m$-dimensional real vector space, and let $e_1,\ldots, e_m$ be a basis such that the $m$-form $f$ has the property $$f(e_1,\ldots,e_m)\neq0$$Is $\operatorname{Alt}(f)\neq0$? Or does ...
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Objects that generalizes Universal enveloping/Clifford/Weil algebra and correspondence to algebraic geometry.

There are evident similarities between Clifford, Weil and Universal Enveloping algebras. Each may be defined directly as a quotient of the tensor algebra $T(V)$ divided by an ideal generated by some ...
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Defining an $A$-algebra structure using a basis and “constants of structure”

Let $A$ be a commutative ring and $E$ an $A$-module with a basis $(e_i)_{i\in I}$. Let $(\alpha_{ijk})_{(i,j,k)\in I\times I\times I}$ be a family of elements of $A$ such that, for $i,j\in I$, the ...
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The variety $\mathcal{S}(v_1 \otimes v_2 \ldots \otimes v_N)$ in $S^N V$

Let $V$ be a $d$-dimensional vector space over $\mathbb{C}$. The space of symmetric tensors $S^N V$ is spanned by the following kind of tensors $$ \mathcal{S}(v_1, v_2, \ldots , v_N) = \sum_{P\in S_N} ...
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Is there a one-to-one correspondence between vector-valued differential forms and real-valued forms?

One way of thinking about vector $V$-differential forms for some real, finite-dimensional vector space $V$ is as alternating and $C^{\infty}(\mathcal{M},\mathbb{R})$-linear functions of the type $$\...
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32 views

Clarification of pseudo-Euclidean space definition

In the definition of pseudo-Euclidean space, we have $\mathbb{R}^{p+q}$ with a non-degenerate quadratic form $Q$ where $p,q\geq 0$, $n\geq 1$ and $p+q=n$. For $x \in \mathbb{R}^{p+q}$ we have $Q(x_1,\...
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1answer
75 views

Is this exterior algebra identity correct? (Repost)

I’m reading DJH Garling’s Clifford Algebras: An Introduction. He defines creation and annihilation operators as follows. Given a finite dimensional vector space $E$ (over $\mathbb{R}$, say), let $A^k(...
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74 views

Are $(1, 0)$ tensors always vectors? (resolved)

An $(r, s)$ tensor $T$ is defined to be an element of the tensor product of a vector space and its dual: $$T \in T^r_sV := V^{\otimes r}\otimes V^{* \otimes s}.$$ However, when $V$ is finite ...
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9 views

Single-factor CP decomposition optimization equivalency with block coordinate descent using the tensor power method

I am trying to understand the objective functions for CP decompositions using the tensor power method as discussed in this paper. In this approach, a series of rank-1 approximations are made using the ...
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1answer
21 views

Clifford map is injective for $B=0$.

Here's my definition of a Clifford algebra: Definition: Let $B(\cdot,\cdot)$ be a symmetric bilinear form on a vector space $V$ over $\mathbb{K}$ and $Q$ its associated quadratic form. The Clifford ...
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27 views

Finding the spectra of difficult parametric matrices

Preparing for entrance exams, I am in need of finding the spectra of the following matrices the most effectively. Anyone up for helping me find the best ways? Matrix 1 | Matrix 2 | Matrix 3 The ...
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15 views

extrior product of a square matrix

I want to understand how one can find the exterior product of square matrices. I searched about it and I find a lot of complicated formulas and without any examples. On the other hand, I found this ...
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1answer
47 views

Correct definition of scalar triple product

Consider orieneted euclidean space $\mathbb{R}^3$. A vector $v\in\mathbb{R}^3$ determines a $1$-form $\omega_v^1$ where $\omega_v^1(w)=\langle v,w\rangle$, where the brackets denotes the equipped ...
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27 views

The optimal solution of tensor Tucker decomposition problem

I have a question about tensor Tucker deocomposition. Recall that in Higher-Order Singular Value Decomposition (HOSVD), each factor matrices $U_i$ is obtained by taking the top $R_i$ left singular ...
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131 views

Generalizing the geometric interpretation of dot product to simple $k$-vectors

Background: For $u, v \in \mathbb R^n$, the dot product $u \cdot v$ can be interpreted geometrically as follows: Its magnitude is the product of the lengths of $u$ and $\operatorname{proj}_{u} v$. ...
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37 views

unique factorization of product of linear functionals

Let $V$ be a vector space over the complex numbers. Let $f_{1\leq i \leq n}$ and $g_{1\leq i \leq n}$ be two sets of nonzero linear functionals on $V$. Suppose we have $$f_1(v) f_2 (v)\ldots f_n (v) = ...
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28 views

Exterior power of totally nonnegative square matrices

An $n$-by-$n$ square matrix $M$ is totally non-negative (TNN) if all its minors are non-negative. If we regard $M$ as the matrix for some linear operator $\varphi:V\rightarrow V$ under some basis $\{...
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107 views

Harder proof using tools from Category Theory rather than Set Theory

For this question I'm assuming a person who knows both basic set theory and category theory (functors, natural transformations, limits, colimits, adjacents). There are theorem proofs that get "...
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21 views

Linear basis of the geometric algebra $\mathcal{G}_{2}$

My question is about the last part of the proof (in bold), for more context, I'll quote all the proof: (Linear basis of $\mathcal{G}_{2}$ ) The elements $\langle 1, e_{1}, e_{2}, e_{1} e_{2} \rangle \...
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Surjectivity of a linear map over an infinite dimensional tensor product

I'd really some help on this problem i've been working on for while. I've got two sets $X$ and $Y$, the map $\overline{\phi}:\mathbb{R}^X \times \mathbb{R}^Y \rightarrow \mathbb{R}^{X\times Y}$ such ...
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63 views

Double Hodge star property

Let $\star$ be the Hodge star operator. Prove that $$\star \star \omega \enspace = \enspace (-1)^{p(n-p)} \cdot \omega$$ for some $\omega \in \Lambda^p$. What I have so far is the following: Without ...
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Prove that $\big[ι_{\vec v}(T)\big](\vec v_1,…,\vec v_{k-1}):=\sum_{r=1}^k(-1)^{r-1}T(\vec v_1,…,\vec v_{r-1},\vec v,\vec v_r,…,\vec v_{k-1})$

Definition Let $V$ be a vector space and $k$ a non-negative integer. Given $T\in\mathcal L^k(V)$ and $\vec v\in V$ let $i_{\vec v}T$ be the $(k-1)$-tensor which takes the value $$ \big[\iota_{\vec v}(...
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28 views

Representing $n$-linear functions as some sort of array?

Let $T$ be a linear transformation between vector spaces $$T: V \rightarrow W$$ Once we select a basis for $V$ and $W$, we can represent $T$ by a matrix. Given a bilinear functional $$B:V \times W \...
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34 views

Multilinear Forms as a Vector Space

if we want to prove that the collection of all multilinear forms is a vector space over $F$, I am having some trouble wrapping my head around some fundamental concepts. By definition, for some $f:V^k\...
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Relationship between the Pfaffian and determinant from exterior algebra

I have been trying to prove the well-known relation linking the Pfaffian and the determinant from their definitions in terms of the exterior algebra, but unfortunately I haven't been able to work it ...
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47 views

Patterns of zeros in nonsingular $0,1$ matrices

I am reviving a question that the OP, Svyashennik Sanya, deleted. Since it will not be visible to all users, I restate it here. It is true that a non-singular $n\times n$ real matrix, all of whose ...
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49 views

$T(M \oplus N) = T(M) \otimes T(N) \otimes T(M) \otimes T(N)…$

Let $R$ be a commutative ring with unity, $M,N$ are $R$-modules, $T(M),T(N)$ their tensor algebras over $R$. In page $571$, D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, the ...
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44 views

What is the pullback of an alternating multilinear map

I'm trying to understand the proof of the statement which says that up to multiplication by a nonzero scalar, there is a unique left invariant volume form on a lie group G. What I don't understand in ...
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1answer
29 views

Are alternating and anticommutative forms equivalent?

A form is defined to be alternating iff having two equal arguments means it is equal to 0. It is defined to be anticommutative iff permuting its arguments means multiplying by the sign of the ...
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32 views

Why does the alternatization of a $k$-linear map indeed produce an alternating $k$-linear map?

The alternating operator $A$ produces for any $k$-linear map $f$ an alternating $k$-linear map $Af$ (the alternatization of $f$): $$Af(v_1, \ldots, v_k) = \sum_{\sigma \in S_k} \text{sgn}(\sigma)f\...
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36 views

Applying a $3$-vector to a vector

Let $V$ be a vector space and $v,w\in V$ two vectors. Let $M$ be an endomorphism of $V$. Then $M(v)\in V$ and $M(v)\wedge v\wedge w\in\bigwedge^3 V$. What does the expression $$(M(v)\wedge v\wedge w)v$...
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1answer
61 views

Decomposition of Linear Maps between Tensor Product Spaces into Linear Maps between the Individual Spaces

What are the sufficient and necessary conditions for representing every linear map $\varphi \in L(V_1\otimes\cdots\otimes V_k; W_1\otimes\cdots\otimes W_k)$ by a tensor in $L(V_1;W_1) \otimes\cdots\...
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1answer
46 views

Specific example of the universal property of $V \otimes W$

I'm trying to familiarize myself with the definition of tensors, so I was wondering if I understood the definition in terms of the universal property. Consider a bilinear $B : V \times W \rightarrow U$...
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27 views

Understanding domain and range of a Tensor as an operator

I am trying to understand the definition of a tensor. The text-book definition of a tensor is a map from $ V \otimes V \otimes V ... \text{(p times)} \otimes V^* \otimes V^* \otimes V^* ... \text{(q ...
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85 views

Show that $V\otimes V\simeq L(V^*,V^*,\mathbb{R})$

I am trying to understand tensor products and I would like to show that $V\otimes V\simeq L(V^*,V^*,\mathbb{R})$. In Lee's book about smooth manifolds is the following proof for the case $V^*\otimes V^...

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