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Questions tagged [multilinear-algebra]

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

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Grounding the concept of a Free Vector space of the cartesian product of two vector spaces

$\def\tv{\tilde{v}}$ $\def\tw{\tilde{w}}$ $\def\F{\mathbb{F}}$ In constructing the Tensor Product of two, finite-dimensional, vector spaces $V,W$ over a field $\F$ it is common to start from the Free ...
Ted Black's user avatar
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$(p, q)$ tensors and multidimensional arrays

I am trying to understand connections between different interpretations of tensors. In many contexts, tensors are treated simply as multidimensional arrays. Let us consider the following example. Let $...
mathslover's user avatar
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Definition of tensor product seems to contradict universal property

$\def\vc#1{\vec{\mathbf{#1}}}$ $\def\cv#1{\tilde{\mathbf{#1}}}$ $\def\qty#1{\left(#1\right)}$ $\def\F{\mathbb{F}}$ In a number of standard textbooks on tensors for physics students (e.g. Tensors: The ...
Ted Black's user avatar
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A countable tensor product: $\ldots \otimes M_{-1} \otimes M_0 \otimes M_1 \otimes \ldots$

It is well-known that if $R, S, T$ are rings, $A$ is an $(R, S)$-bimodule and $B$ is a $(S, T)$-bimodule, we can form the $(R, T)$-bimodule $A \otimes_S B$ as the quotient of $A \otimes_\mathbb{Z} B$ ...
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Constructing a basis for a tensor product

Let $V$ and $W$ be vector spaces of a field $\mathbb{K}$. Let $\left\{v_i : i \in \mathcal{I}\right\}$ be a basis of $V$, and $\left\{w_j : j \in \mathcal{J}\right\}$ be a basis of $W$. Let $V \otimes ...
mathslover's user avatar
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Werner Greub's formulation of the Universal Property of the Tensor Product

$\def\id{\operatorname{id}}$ $\def\Im{\operatorname{Im}} $In Section 1.4 of Multilinear Algebra Werner Greub starts with a bilinear map $\otimes: E \times F \rightarrow T$ where $E,F,T$ are vector ...
Ted Black's user avatar
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Equivalent definitions of tensor power of a vector space

I have two definitions of the tensor power $T^nV$ of a vector space $V\in \bf{kVect}$. For every $n$-multilinear map $f:V\times ...\times V\rightarrow W$, there exists a unique $\bar{f}:T^nV\...
Wyatt Kuehster's user avatar
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About Hom (TM ,Hom (TM,ν))≅ Hom (TM⊗TM,ν)

In the book ," Characteristic Classes " by J.W.Milnor and J.D.Stasheff there is a problem ( 5-B) in which the following isomorphism had been mentioned: Hom ($TM$ ,Hom ($TM ,\nu )) \cong $ ...
Math Learner's user avatar
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Nice proof that $\text{Alt}$ is natural.

There is a pair of functors $T:\text{kVect}\rightarrow \text{kAlg}$ and $\Lambda:\text{kVect}\rightarrow \text{kAlg}^-$ which are left adjoints to the forgetful functors $U$ (forget the multiplication ...
Wyatt Kuehster's user avatar
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Sign of the permutation when I show that $\ast \ast w= (-1)^{n(n-k)} w$ for the Hodge operator

Let $w=\sum_{I} a_{I}dx_{I}$ be a $k$-form in $\Bbb R ^n$. Let us consider the Hodge operator in a combinatorial form, i.e. as an $(n-k)$ form such that $$\ast(dx_{i_{1}} \wedge \cdots \wedge dx_{i_{...
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What if a CP/PARAFAC tensor decomposition can be further decomposed, "recursively"?

Standard CP/PARAFAC decomposition: A tensor $\mathcal{T}$ in shape $(I_1,\dots,I_N)$ is produced by $N$ matrices $\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}$ where each $\mathbf{A}^{(n)}$ is in shape $(...
graphitump's user avatar
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Exterior and symmetric powers without choice

Let $R$ be a commutative ring, $F$ a free $R$-Module and $n\in \mathbb{N}$. Can it be proven in ZF that the canonical projections $F^{\otimes n}\twoheadrightarrow \bigwedge^n(F)$ and $F^{\otimes n}\...
Lucina's user avatar
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Expansion of a p-form in basis one-forms - antisymmetry of coefficients

I have found the following equation for the expansion of a general p-form in a book: $\phi = \sum_{i_1<i_2<...<i_p}\phi_{i_1,...,i_p}\sum_{P\in S_p}sgn(P) e^{i_{P1}} \otimes e^{i_{P1}} \...
Takitoli's user avatar
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Can we construct the exterior algebra just from simple multivectors?

$ \newcommand\K{\mathbb K} \newcommand\Ext{\mathop{\textstyle\bigwedge}} \newcommand\Lip{\mathrm{Lip}} \newcommand\ev{\mathrm{ev}} \newcommand\Gr{\mathrm{Gr}} $Let $V$ be a finite-dimensional $\K$-...
Nicholas Todoroff's user avatar
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1 answer
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Show that the dual Lefschetz operator applied to a two-form $\alpha$ is explicitly given by $\Lambda \alpha =\sum_i \alpha(x_i, y_i)$.

Choose an orthonormal basis $x_1, y_1 = J(x_1), \dots , x_n, y_n = J(x_n)$ of an euclidian vector space $V$ endowed with a compatible almost complex structure $J$. Show that the dual Lefschetz ...
Rene's user avatar
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Dimension of Kernel for Linear Map on Tensor Product Space

Let $U = \mathbb{R}^{d_1 \times d_2}$ and $V = \mathbb{R}^{d_2 \times d_3}$ be spaces of matrices. Consider the linear map: $$T: U \otimes V \to \mathbb{R}^{d_1 \times d_3}$$ defined by $$T(A \otimes ...
Alex's user avatar
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How are polynomials of tempered distributions defined?

I am reading a textbook on rigorous quantum mechanics and quantum field theory in which there often appears statements such as Let $A(\phi)$ be a polynomial function defined on $\mathcal{S}'(\mathbb{...
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Clarification needed on the multiplication of elements of a tensor algebra

I'm reading Rotman's Advanced Modern Algebra: Part 1 and which gives the definition of a tensor algebra as: If $M$ is a $k$-module, define $$ T(M)=\bigoplus_{p \geq 0}\left(\bigotimes^p M\right)=k \...
ctk's user avatar
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What operation on matrices corresponds to the curl of a vector field?

Given the total derivative $Df$ of a (sufficiently) smooth function $f:\mathbb{R}^n \to \mathbb{R}^n$, the trace of the total derivative matrix corresponds to the divergence of $f$ (considered as a ...
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If $T \in \mathscr L(V, W)$, then there exists a map $T^∗: \tau^k(W) \to \tau ^k(V ).$

$\mathscr L(V, W):=$ space of all linear transformations from $V$ to $W.$ $\tau^k(W):=$ Space of all $k-$linear transformation from $W\times W ...\times W(k-\text{times})\to \mathbb R.$ Similarly, $\...
Unknown x's user avatar
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Two approaches to tensors?

This is a soft question so I'm not completely sure if it is appropriate here, but anyway. Lately I have been studying tensors as I'm working with tensor-machine-learning. Initially I used standard ...
AyamGorengPedes's user avatar
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What is the simple reason why there is no simple solution to a system of quadratic equations? [duplicate]

Let's consider a system of $n$ equations and $n$ real unknowns $(x_1, \dots, x_n)$: $$\sum_{k=1}^n A_{ik} x_k = B_i, \qquad \text{with } i\in\{1,\dots,n\}$$ The solution of which exists as long as the ...
Davius's user avatar
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3 answers
531 views

What is the empty tensor product of vector spaces?

The tensor product of a space with itself once is $V^{\otimes1}$, but what is $V^{\otimes0}$? Since it is an empty tensor product, it is - a fortiori - an empty product. So I'm looking for a "$1$&...
Hank's user avatar
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Values of differential 2-forms on $k$-dimensional planes

I've found this demonstration (for this problem taken from "Mathematical methods of Classical Mechanics" by V. I. Arnol'd), and I could not decode this particular step: A $k$-dimensional ...
Lo Scrondo's user avatar
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Wedge of vectors and wedge of forms

Consider the $\mathbb{R}^3$ vectors expressed on terms of the canonical basis $$X=\sum_{i=1}^3x_ie_i,\,Y=\sum_{i=1}^3 y_ie_i,$$ so the wedge product of vectors is $$X\wedge Y=(x_2y_3-x_3y_2)e_1-(...
Gonzalo de Ulloa's user avatar
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2 answers
200 views

Inversion of a matrix equation

Is there a general way to invert (solve for $u$) this? $$\sum_{ij}R_{ijk}a_iu_j = -x_k$$ With $a,u,x \in \mathbb{R}^N$. $R_{ijk}$ is symmetric in the last two indices. So really I'm trying to invert ...
Gennaro Marco Devincenzis's user avatar
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1 answer
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Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
Brendan Murphy's user avatar
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Induced change of basis on a (p,q) tensor

I'm struggling to simplify the last step of a $(p,q)$ tensor and how its components change with a linear change of basis on the associated vector space. So far I have: Given a vector space $V$ over ...
Tyler Roche's user avatar
2 votes
1 answer
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Tensor confusion: scalar $\otimes$ vector = OK?...

I am trying to understand tensors and one particular question have caused me a great deal of confusion. The particular example with the metric tensor below is an attempt to highlight where my ...
evolhart's user avatar
2 votes
1 answer
67 views

Equality of $2$nd compound matrices implies equality up to sign

There’s a bit of multilinear-algebraic folklore knocking around in my head that I’d like to cite and use but that I can’t track down for the life of me: Let $n > 2$ be an integer, and let $a,b \in ...
Branimir Ćaćić's user avatar
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Why is this minimizer obvious?

Lately I have been studying multi-linear algebra, and I'm trying to build my geometric intuition on it. A problem I'm currently faced with is to maximize $$\max_U <U, YX^T>_F$$ subject to ...
AyamGorengPedes's user avatar
5 votes
1 answer
81 views

Kernel of the action of GL(V) on exterior square of V

I wonder whether anyone knows a reference for the following result? I can give a shortish proof, but would prefer to cite the literature if possible. Theorem Let $V=F^n$ be an $n$-dimensional $F$-...
Glasby's user avatar
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$\mathbb{C}-$linear extension of 2-forms to (1,1)-forms

I am trying to analyze a bit how we can extend a differential form to the complexification $V\otimes\mathbb{C}=V_{\mathbb{C}}$ of the vector space. Of course, you can do this for a general $k$-form, ...
領域展開's user avatar
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1 vote
1 answer
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Universal metric of a bivariant tensor $t^i_j$.

I'm reading Elements for Physics by Albert Tarantola, and get stuck on page 16. Before this page, the author defined a metric on a linear space $S$ as a map to its dual $\mathbf{G} : S \to S^*$ that ...
Mugenen's user avatar
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Tensor product of $n$ modules with minimal assumptions

Let $R_1,\ldots, R_n$ be (commutative) rings, and $M_1 , \ldots, M_n$ be modules. What are the minimal assumpsions about the rings and modules needed in order to define a tensor product $M_1 \otimes_{...
Robert's user avatar
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Tensor product of linear forms.

Let $P$ be a module over a conmutative unitary ring $A$ and $\mathcal{J}^r(P)$ be the module of $r$-linear forms $P \times \cdots \times P \longrightarrow A.$ If $\alpha \in \mathcal{J}^r(P)$ and $\...
Luis Esquivias's user avatar
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1 answer
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Fundamental form $\omega=\sum_{i\leq m}v^*_i\wedge (Jv_i)^* $ with a complex structure $J$

Let $V$ be a $\mathbb{C}-$ vector space, $J$ an almost complex structure on $V$ and take a real orthonormal basis $\langle v_1,Jv_1,\ldots,v_n,Jv_n\rangle $ with a scalar product $\langle,\rangle = \...
領域展開's user avatar
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Exterior algebra decomposition natural projections $ Π^{p,q}:\bigwedge {V}^*_{\mathbb{C}} \to \bigwedge^{p,q}V$ (Huybrechts book)

Let $\bigwedge^{p,q}V:=\bigwedge^{p}V^{1,0}\wedge \bigwedge^{p}V^{0,1}$ and $\bigwedge {V}^*_{\mathbb{C}}=\bigoplus_k \bigwedge^{k} V_{\mathbb{C}} $. Then one defines the natural projection $$ Π^{p,q}...
領域展開's user avatar
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What does it mean, The space $L_n (V^n;K)$ of alternating n-linear forms is of dimension one?

Reading about the fundamental theorem of alternating applications which says Given 2 vector spaces over $K$, $(V;K)$ and $(W;K)$. If $dim\ \ V=n$ and a base of V is {$u_1...u_n$} I saw that there is ...
MonkeyDL's user avatar
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63 views

Symmetrization and antisymmetrization of multilinear transformations

Let $\beta: \Bbb R^n \times \dots \times \Bbb R^n \to \Bbb R^m$ be $r$-linear. Define $$ \operatorname{symm}(\beta)(v_{1},\dots,v_{r}) = \frac{1}{r!} \sum_{\pi}^{}\beta(v_{\pi(1)},\dots,v_{\pi(r)})$$ ...
Anna's user avatar
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I have confusion understanding why Zero Vector space has dimension zero? [duplicate]

Here's what I understand from Basis. Basis: it's set of linearly independent vectors which can span the vector space. Basis for Zero vector space: case 1: when { 0 } , it's singleton set , since ...
Inception's user avatar
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102 views

How to deduce: $T(x_1, \dots , x_i+x_j, \dots, x_i+x_j, \dots , x_n) = 0$?

studying about alternating multilinear applications I came across this expression. I understand that it represents the demonstration that an application T is antisymmetric (since it changes sign when ...
MonkeyDL's user avatar
1 vote
1 answer
71 views

How/why does definition of eigenvalue/vector implicitly require a choice of inner product?

I forget exactly where, but I've heard/read multiple times it be mentioned that defining eigenvalues/vectors implicitly requires a choice of inner product. Question: Is it more correct to say that &...
hasManyStupidQuestions's user avatar
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Help me solve this question if not atleast tell me What are the backgrounds needed to study to solve this question about Kenmotsu Manifold?

So how do I solve this 𝐵 (𝑋, 𝑌)𝑍=𝑅 (𝑋, 𝑌)𝑍 + 1 𝑛+3 × [𝑔 (𝑋, 𝑍) 𝑄𝑌 − 𝑆 (𝑌, 𝑍) 𝑋 − 𝑔 (𝑌, 𝑍) 𝑄𝑋 + 𝑆 (𝑋, 𝑍) 𝑌 𝑔 (𝜙𝑋, 𝑍) 𝑄𝜙𝑌 − 𝑆 (𝜙𝑌, 𝑍) 𝜙𝑋 − 𝑔 (𝜙𝑌, 𝑍) 𝑄𝜙𝑋 + ...
ThematicalMathee's user avatar
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The Riemann Curvature of The Round Sphere

Note first the following definition of the Riemann curvature tensor I have been using: $$\text{Riem}(\omega, Z, X, Y) := \omega\ (\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z - \nabla_{[X,Y]}Z)$$ where ...
Taylor Rendon's user avatar
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Does the Hessian correspond to the exterior derivative of the gradient 1-form? Or does its skew-symmetrization?

Question: Given a twice totally differentiable (not necessarily $C^2$) function $f: \mathbb{R}^m \to \mathbb{R}^n$, do its $n$ Hessian matrices correspond to the exterior derivatives of its $n$ ...
hasManyStupidQuestions's user avatar
1 vote
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65 views

Property of the alternating operator

Let $\Lambda^kV^*$ be the space of alternating k-linear forms on $V^k:=V \times \dots \times V$. Such that for $\omega \in \Lambda^kV^*$, $$\omega(v_1, \dots, v_i+v_j,\dots, v_k)=\omega(v_1, \dots, ...
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1 answer
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On the tensor product

Let us start with $V,W$ to $R$-modules. In order to define their tensor product we first introduce something called the free R-module $Free(V \times W)$ whose basis is given by the set of all ordered ...
user57's user avatar
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What is the different meaning of $k$-tensor and tensor of type $(p,q)$?

Let $V$ be a $n$ dimensional vector space over $\mathbb R$ and $k,p,q\in\mathbb N$. A $k$-tensor is a $k$-multilinear functional on $V$, that is a map \begin{align} f:&& V^k&\...
PermQi's user avatar
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2 votes
1 answer
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About $\operatorname{Alt}(\varphi_{i_1}\otimes\cdots\otimes\varphi_{i_k})$ ("Calculus on Manifolds" by Michael Spivak)

I am reading "Calculus on Manifolds" by Michael Spivak. The author wrote as follows: Since each $\operatorname{Alt}(\varphi_{i_1}\otimes\cdots\otimes\varphi_{i_k})$ is a constant times one ...
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