Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

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44 views

Coalgebras & Coideals: Why does $ker(\pi \otimes id_C ) = I\otimes C$ hold?

In a proof on comodules and coideals I found the following passage: “Let $C$ be a coalgebra, and $I \subset C$ a vector subspace. Let $\pi$: $C \rightarrow C/I $ be the canonical projection. Consider ...
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1answer
37 views

Tensor product of two $n\times n$ orthogonal matrices with determinant $+1$

If two square matrices A and B are two $n\times n$ orthogonal matrices with determinant unity i.e., $\det A=\det B=+1$ and $A^TA=B^TB=I$, will the tensor product $C\equiv A\otimes B$ also be ...
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1answer
36 views

$M \otimes_R N \cong M \otimes_S N$ as $S$-modules?

Let $R$ be a commutative ring with identity, and let $S \subseteq R$ be a subring of $R$, sharing a common identity. Moreover, let $M$ and $N$ be $R$-modules. Is it then true that $M \otimes_R N \cong ...
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1answer
51 views

Kernel of a morphism between projective modules and base change

Let $A$ be a ring (Noetherian if it helps) and $f : M \to N$ be a morphism between finite projective $A$-modules. Let $B$ be an $A$-algebra. Is it true that if $f\otimes_A B$ is injective then $\ker(...
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24 views

Proof that the exterior power fulfils an universal property using tensor product

before I ask my question, I briefly review two definition which I use: (1) Tensor product of vector spaces: Let $V_{1},\dots,V_{n}$ be vector spaces over the same field $\mathbb{F}$. Then the ''...
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1answer
26 views

The proof of a tensor cross product identity

If $l=a\times b,l_i=\epsilon_{ijk}a^jb^k$. then $a^ib^j+a^jb^i=l^il^j-l^2g^{ij}$. $g$ is the metric tensor. I tried to dot product $g_{ij}$ to two sides, then I found it became $2(a\cdot b)^2=2(a\cdot ...
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1answer
62 views

Tensor product raised to power $N$.

Can the following quantity be reduced to further $[A \otimes B + (\mathbb{1} - A) \otimes D]^N$, for positive interger $N$? Here, $\otimes$ denotes the Kronecker (tensor) product and the matrices $A$ ...
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23 views

Adjunction property for the frobenius push-forward

I am reading the paper and I have an issue with the last part concerning the adjunction. Denote $T_{X|S}=Der(\mathcal O_X,\mathcal O_X)$ with $X$ a scheme and let $\omega_{X|S}$ be a line bundle on $...
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18 views

Does quadruple product identity holds for any metric tensor?

$(A\times B)\cdot(C\times D)=(A\cdot C)(B\cdot D)-(A\cdot D)(B\cdot C)$ If I take any metric tensor $g_{ij}$ instead of $\delta_{ij}$, will this identity still holds? I found as if $g$ is a symmetric ...
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1answer
55 views

How are the following two rings isomorphic?

The paper here in Construction 4.16 makes the following claim that I'm unable to unpack (though my question should be self-contained here): Let $R$ be a commutative ring, and let $r\in R$ be an ...
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30 views

two structures of $\mathcal O_X$-Algebra for the sheaf of principal parts

Let $X \to S$ is be a morphism of schemes, and $F$ an $\mathcal O_X$-modules. Let $P=\mathcal O_X\otimes_{f^{-1}(\mathcal O_S)}\mathcal O_X$ I don't understand the following two structures of $\...
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1answer
59 views

The proof of this tensor identity

I found that $\epsilon^{ijk}g_{jm}g_{kn}=\det(g)g^{il}\epsilon_{lmn}$, is it correct? What's proof of it? This formula was found when I was studying tensor cross product, I tried to prove that if $A\...
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1answer
57 views

About two definitions of tensor product. (in James R. Munkres's book and Ichiro Satake's book)

I am reading "Analysis on Manifolds" by James R. Munkres. Definition: Let $f$ be a $k$-tensor on $V$ and let $g$ be an $l$-tensor on $V$. We define a $k+l$ tensor $f \otimes g$ on $V$ by the ...
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1answer
13 views

Tensor of s.e.s. of projective modules [closed]

Let $f:R \rightarrow S$ be a ring homomorphism. All modules are right R-modules.Let $\otimes_R S:\mathbb{P}(R) \rightarrow \mathbb{P}(S)$ be the functor sending a projective $R$-module $P$ to the ...
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2answers
56 views

Finitely generated module over PID with tensor with itself is zero

Hello I have the next doubt about this problem: Show that if $A$ is a finitely generated module over a PID and $A\otimes_{\Lambda}A=0$, then $A=0$. I have done the next thing, I consider the next ...
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1answer
33 views

Existence of the tensor product

Let $E_1$ and $E_2$ be $\mathbb{K}$-vector spaces. Lets also consider the set $F$ the functions $\xi:E_1\times E_2\to \mathbb{K}$ such that \begin{equation} |\{(\mathbf{u},\mathbf{v})\in E_1\times E_2:...
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1answer
31 views

$\mathbb{Z}[T]$-module and extension

I consider the structure of $\mathbb{Z}[T]$-module on $\mathbb{Z}$ given by the multiplication $P \times a := P(0) \times a$. Now i consider a morphism of ring $\phi : \mathbb{Z}[T] \to R$, that ...
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1answer
105 views

Isomorphism concerning tensor product

While reading algebraic number theory, I came across the following fact which I don't know how to prove. Let $A$ be an integral domain with field of fraction $K$ and $ L/K$ is a finite separable ...
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2answers
82 views

Derivative of Frobenius norm of tensor product in component-free notation

I need to take a derivate with respect to $w$ of the following function: $$ f(x,w)=\left\lVert x \otimes w\right\rVert _{F}, $$ where $x \in \mathbb{R^{n}}$, $w \in \mathbb{R^{n}}$, $\otimes$ is a ...
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Is this tensor graph product known?

I have a question about graph products. Imagine that I have a graph with adjacency matrix A. I want to build a product graph with adjacency matrix G such that: $G_{(ik), (jl)} = A_{ij}A_{kl}\times (1-...
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1answer
56 views

Trace norm sum of tensor products

For two arbitrary matrices $A, B$, it is fulfilled $\|A+B\|\leq\|A\|+\|B\|$, where $\|\cdot\|$ is the trace norm. Is the following satisfied: $\|A\otimes1+1\otimes B\|=\|A\|+\|B\|$? As from the ...
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1answer
34 views

Induced representation of the symmetric group and tensor product of it

Given $n_1,n_2,n_3$ be positive integers, let $V_1,V_2,V_3$ be representations of $S_{n_1}, S_{n_2}, S_{n_3}$, respectively. Here, $S_n$ is the symmetric group on $n$ letters. I wonder that $$ (( V_1 ...
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1answer
49 views

Tensor product of suspensions and its inverse

Let $A$ be a graded vector space. Define the suspension $S(A)^d=A^{d+1}$, where the superindex indicates the degree component, so that the map $S:A\to S(A)$ given by the identity in each degree has ...
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48 views

is a Block matrix a Tensor?

Currently I am starting to study tensor calculus and I came across the definition of the tensor product, and more specifically the definition of tensor rank (ex. a tensor product of 2 rank 1 tensors (...
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1answer
49 views

What is the rank of $\mathbb{Z}[i] \otimes_{\mathbb{Z}[i]} \mathbb{Z}[i]$ as $\mathbb{Z}$-module?

What is the rank of $P=\mathbb{Z}[i] \otimes_{\mathbb{Z}[i]} \mathbb{Z}[i]$ as $\mathbb{Z}$-module ? Also I have to provide a basis. My guess is the following. As a $\mathbb{Z}[i]$ module clearly $P \...
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0answers
73 views

What are the matricization and vectorization of tensor products?

I'm trying to understand the concepts of matricization (matrix unfolding) and vectorization of tensor products. In the past, I've only dealt with tensor products of infinite-dimensional Banach and ...
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1answer
15 views

Computing the basis of $\mathbb{T}_{2}\left(\mathcal{P}_{2}(\mathbb{R})\right)$ associated with the canonical basis in $\mathbb{R}^{3}$

I am looking at the 2-covariant tensor space over the field of polynomials of degree equal or less than 2, $\mathbb{T}_{2}\left(\mathcal{P}_{2}(\mathbb{R})\right)$. (Sorry if this notation is not ...
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1answer
34 views

Relation between products of groups and matrix representation?

I have done a little bit of abstract algebra with groups. Most of it relates to rather practically oriented applications in engineering like rotation groups, permutations, affine transformations. For ...
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12 views

Operations in tensor algebra: unfolding and contraction

For my study the tensor algebra I am using the following two papers: 1 [Era of Big Data Processing by Andrzej CICHOCKI]1 2 [Tensor Decompositions and Applications by T KOlda]2 My questions: Tensor ...
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1answer
38 views

Isomorphism between $K_\mathbb{R}$ and $K\otimes_{\mathbb{Q}}\mathbb{R}$

I am reading Neukirch's Algebraic Number theory and I am a little bit confused about the part where he mentioned that $K_\mathbb{R}$ and $K\otimes_{\mathbb{Q}}\mathbb{R}$ are isomorphic via the map $\...
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1answer
80 views

$a\otimes(-a) = a \otimes (1-a) + a^{-1}\otimes(1-a^{-1})$???

J. Browkin writes in his article "K-Theory, Cyclotomic Equations, and Clausen's Function" (which appears on Chapter 11 of Lewin's book "Structural properties of polylogarithms") that for any field $F$ ...
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2answers
64 views

Are inner products always dominated by the norm of tensor products?

Prove or disprove the following possible generalization of Cauchy-Schwarz inequality: Let $H$ be a Hilbert space and let $A_1, \dots A_n, B_1, \dots B_n \in H$. Then $|\sum_{i=1}^{n} \langle A_i, B_i ...
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1answer
77 views

Infinite tensor product of vector spaces as direct limit of finite families of vector spaces

I was looking for the infinite tensor product of vector spaces and in literature (for instance, Atiyah and Macdonald's book), I found it for algebras as the direct limit of finite families of algebras....
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1answer
53 views

Two notions of a tensor

The way I've been learning tensors—building towards differential forms—has the tensor product for vectors $u,v\in V$ (say, finite-dimensional and over some field $\mathbb F$) as $$u\otimes v=[(u,v)]\...
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1answer
21 views

A quotient of tensor algebra of $V \otimes k$

Let $k$ be a field and $V$ a $k$-vector space. Denote the inclusion of $k$ in $k \oplus V$ by $m$. A book that I'm consulting (Ramanan's Global Calculus) asks the reader to prove that the quotient of ...
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23 views

Why into Minkowski metric we use a pseudo-Riemannian metric and not, simply, a pseudo-metric?

A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space $R^n$ A pseudometric space $(X,d)$ is a set $X$ together with a non-negative real-valued function $d\colon X \...
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1answer
83 views

$\mathbb{Z}[G^{n+1}] \otimes_{\mathbb{Z}[G]} \mathbb{Z} \cong \mathbb{Z}[G^n]$ as $\mathbb{Z}$-module

Let $G$ be a group. If $M$ is any right $G$-module, then we can consider $M$ as left $G$-module also under the action $g.m:= mg^{-1}$, where $m \in M$ and $g \in G$. Consider $\mathbb{Z}[G^{n+1}]$ as ...
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1answer
51 views

What is the submodule of $M \otimes_R N$ generated by $1 \otimes_R n$?

For some context, I am trying to prove that extension and restriction of scalars are an adjoint pair. Formally, given a commutative ring map $f: A \rightarrow B$ and $M$ an $A$-module, $N$ a $B$-...
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26 views

Different types of tensor products of two vectors?

Consider two vectors such as $$v_1=(a,b,c)~~~,~~~v_2=(d,e,f)$$ I am wondering how many different useful ways to take a tensor product of these two vectors exist such that a matrix is produced? For ...
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1answer
46 views

Expressing Volume Form in local coordinates on Riemannian Manifold

I'm just beginning to study Riemannian Geometry, and in particular the volume form on a Riemannian Manifold $(M, g)$. It was first introduced to me as a differential $n$-form $dV$ for which $dV(e_1, \...
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18 views

Express the power of a dot product with tensor

Given two vectors $\boldsymbol{n}$ and $\boldsymbol{r}$, we know the dot product is $$d=\boldsymbol{n}\cdot \boldsymbol{r}$$ The square of $d$ is $$d^2=(\boldsymbol{n}\cdot \boldsymbol{r})^2 = \...
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40 views

Levi-Civita sign

Can any one explain to me why $$\sum_{k=1}^3 \varepsilon_{ijk}\varepsilon_{kij}=1$$ (Summing on $k$ from 1 to 3) Im trying to understand the index notation but cant see it yet. Thanks.
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Tensor Product of Two Group Representations (Action Well-definedness)

I am studying the basic of representation. Coming back from https://en.wikipedia.org/wiki/Tensor_product_of_representations, I am having trouble filling out details why tensor product of two (finite ...
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1answer
49 views

Free vector spaces and construction of Tensor Product

The whole (intuitive) idea of the necessity of tensor product vector space is almost understood, I mean: given our prior experience with the concept of multiplication (basic elementary one like in ...
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1answer
21 views

Cartesian Components of a Tensor

Show that any tensor, say the third-rank tensor $\textbf{T}$, can be expanded in terms of tensor products of the basis vectors: $$\textbf{T} = T_{ijk} e_i \otimes e_j \otimes e_k$$ and that the ...
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1answer
53 views

Hints to finish proof that $(Q_R/R) \otimes_R (Q_R/R) = 0$?

Here is the work I've done so far. I believe my hang-up is that I'm still learning the basic properties of quotient rings and not so much because of anything about tensor products. I want to say I can ...
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36 views

Terms in shallow water equation in 2D

I am reading pde's models about fluids. In the literature appear the following equations for the case of shallow water models in 2D $$ \frac{\partial H}{\partial t}+\operatorname{div}(H \mathbf{u})=...
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10 views

A tensor operation that results first order tensors

The tensor product of $n$ vectors results a tensor of order $n$ (polyadic tensor). Is there any operation to perform on zero order tensors (scalars) that results first order tensors (vectors)?
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42 views

Exterior power of symmetric product of GL(2,R) tensor representations

A paper I am reading uses an identification of $GL(2,\mathbb{R})$ representations: \begin{align} \Lambda^2(\text{Sym}^3\mathbb{R}^2) = (\text{Sym}^4\mathbb{R}^2 \otimes \Lambda^2\mathbb{R}^2 ) \...
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1answer
78 views

Cholesky decomposition of tensor product

Let $A\in\mathbb R^{n\times n}$, $B\in\mathbb R^{m\times m}$ be symmetric, positive definite, matrices. Let $C = A\otimes B$ be their tensor product. I want to compute the Cholesky decomposition of $C$...

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