Questions tagged [multilinear-algebra]
For questions about the extension of linear algebra to multilinear transformations of vector spaces.
1,113
questions
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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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38 views
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
$$\Phi: \bigwedge\nolimits^2 \mathbb{R}^n \to \...
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42 views
Berger notation for a triple product of tensors
The following definition is given in a paper by M. Berger:
Here, $(M,g)$ is a Riemannian manifold and $S^2(M)$ is the space of symmetric $2$-tensors on $M$. Is there a coordinate-free equivalent for ...
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0answers
24 views
On the solution space of linear matrix equations.
Consider a linear matrix equation
$$ Y = \sum_{i} A_i X B_i^T = \sum_i (B_i\otimes A_i) \cdot X = T\cdot X $$
When does the solution space admit a basis consisting only of rank-1 matrices?
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247 views
Why are Tensors (Vectors of the form a⊗b…⊗z) multilinear maps?
In our linear algebra course we defined the Tensor Product between two vector spaces as an Operation so that this Diagram commutes:
Here $\varphi$ and $\iota$ are multilinear maps and $\psi$ is a ...
3
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0answers
61 views
An svd-like tensor decomposition splitting a tensor into two lower-dimensional tensors and a singular vector
I have also posted this question on mathoverflow, but it seems there are a lot of questions related to SVDs here and a tag "tensor-decomposition", so I will give it a shot.
I am looking for ...
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1answer
60 views
How to find Nullspace of Tensor / Kernel of a multilinear map?
[ Disclaimer: I am only starting to get my head around concepts of tensor calculus, so I apologize in advance for the lack of clarity or asking something trivial. I came across it trying to apply it ...
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13 views
Uniqueness of second -and fourth-order moment tensors of vectors
Let $\{v_1, v_2, v_3\}$ and $\{w_1,w_2,w_3\}$ be two sets of vectors in $\mathbb{R}^2$; we can assume the $v$s are pairwise linearly independent, and likewise for the $w$s.
I'd like to show that if
$$\...
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2answers
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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1answer
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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50 views
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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23 views
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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26 views
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
$$ \mathcal{S}(v_1, v_2, \ldots , v_N) = \sum_{P\in S_N} ...
2
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1answer
54 views
Is there a one-to-one correspondence between vector-valued differential forms and real-valued forms?
One way of thinking about vector $V$-differential forms for some real, finite-dimensional vector space $V$ is as alternating and $C^{\infty}(\mathcal{M},\mathbb{R})$-linear functions of the type
$$\...
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1answer
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(...
3
votes
1answer
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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0answers
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 ...
0
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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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0answers
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 ...
4
votes
1answer
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 $\operatorname{proj}_{u} v$.
...
4
votes
1answer
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
$$f_1(v) f_2 (v)\ldots f_n (v) = ...
0
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1answer
28 views
Exterior power of totally nonnegative square matrices
An $n$-by-$n$ square matrix $M$ is totally non-negative (TNN) if all its minors are non-negative. If we regard $M$ as the matrix for some linear operator $\varphi:V\rightarrow V$ under some basis $\{...
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107 views
Harder proof using tools from Category Theory rather than Set Theory
For this question I'm assuming a person who knows both basic set theory and category theory (functors, natural transformations, limits, colimits, adjacents).
There are theorem proofs that get "...
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0answers
21 views
Linear basis of the geometric algebra $\mathcal{G}_{2}$
My question is about the last part of the proof (in bold), for more context, I'll quote all the proof:
(Linear basis of $\mathcal{G}_{2}$ ) The elements $\langle 1, e_{1}, e_{2}, e_{1} e_{2} \rangle
\...
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0answers
26 views
Surjectivity of a linear map over an infinite dimensional tensor product
I'd really some help on this problem i've been working on for while. I've got two sets $X$ and $Y$, the map $\overline{\phi}:\mathbb{R}^X \times \mathbb{R}^Y \rightarrow \mathbb{R}^{X\times Y}$ such ...
0
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1answer
63 views
Double Hodge star property
Let $\star$ be the Hodge star operator. Prove that
$$\star \star \omega \enspace = \enspace (-1)^{p(n-p)} \cdot \omega$$
for some $\omega \in \Lambda^p$.
What I have so far is the following: Without ...
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0answers
67 views
Prove that $\big[ι_{\vec v}(T)\big](\vec v_1,…,\vec v_{k-1}):=\sum_{r=1}^k(-1)^{r-1}T(\vec v_1,…,\vec v_{r-1},\vec v,\vec v_r,…,\vec v_{k-1})$
Definition
Let $V$ be a vector space and $k$ a non-negative integer. Given $T\in\mathcal L^k(V)$ and $\vec v\in V$ let $i_{\vec v}T$ be the $(k-1)$-tensor which takes the value
$$
\big[\iota_{\vec v}(...
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1answer
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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2answers
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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59 views
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 ...
3
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1answer
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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0answers
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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1answer
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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2answers
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\...
0
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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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0answers
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 ...
5
votes
2answers
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^...
