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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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The action of tensor product over N terms on a ket.

Equation (6) of the paper titled, Multi-player and Multi-choice quantum game has left me puzzled-after many hours-as to how it is being derived. My working begins from the generic form seen just after ...
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1answer
26 views

Tensor product of module homomorphisms is element of tensor product or mapping?

Here: $A$ is a ring. $E,E'$ are right $A$-modules, $F,F'$ are left $A$-modules. $u:E\rightarrow E'$ and $v:F\rightarrow F'$ are $A$-module homomorphisms. With this setup, I didn't understand ...
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1answer
51 views

Example of tensor product of two representations

This is similar to what Serre wrote in his book on linear representations of finite groups by Serre: There is also the tensor product which has the properties of "multiplication". Let $V_1$ and $...
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2answers
39 views

Equivalence between these tensor product definitions

Let $V$ and $W$ be vector spaces. Then the tensor product $V \otimes W$ of $V$ and $W$ is the vector space $V \otimes W$ together with a bilinear map $\phi: V \times W \rightarrow V \otimes W$ such ...
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1answer
27 views

When do tensor products of elements coincide

Let $M,N$ be $R$-modules and $m \otimes n, m' \otimes n' \in M \otimes_R N$ non-zero (EDIT) elements. When does $m \otimes n = m' \otimes n'$ hold? Obviously, this is true if either $(m',n)=(rm,rn')$ ...
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28 views

Tensor product notation as a power

I chance upon a notation while reading a paper which I do not quite understand. Suppose that $\hat{J}$ is a operator in a tensor product of two N$ dimensional Hilbert space. Explicitly, it is given ...
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2answers
37 views

Identifying simple tensors.

Let $S$ be a domain. I want to determine whether or not, every element of $\text{Frac(S)}\otimes_S M$ is a simple tensor, where $M$ is any $S$-module. I couldn't produce a tensor that is not pure in ...
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28 views

No canonical correspondance between bilinear forms $b:V\times V \rightarrow \mathbb{R}$ and linear forms $\hat{b}:V\otimes V \rightarrow \mathbb{R}$?

In the wiki on bilinear forms, the universal product says that to each bilnear map $b : V\times V \rightarrow \mathbb{R}$ we can associate a linear map $\hat{b} : V\otimes V \rightarrow \mathbb{R}$, ...
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39 views

Injectivity of the usual map $C(X)\otimes C(Y)\to C(X\times Y)$.

Let $X$ and $Y$ be compact Hausdorff spaces and consider $C(X)$ and $C(Y)$ the spaces of real or complex-valued continuous functiosn on $X$ and $Y$ respectively. It is "well-known" that $C(X\times Y)$...
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1answer
37 views

Tensor matrix product invariant under rotation matrix

$\alpha$ is a tensor and its multiplication with a matrix $M\in \mathbb{R}^{3x3}$ is $$(\alpha (M))_{ij} = \sum_{k,l} \alpha_{kl,ij} M_{kl}$$ And its multiplication is invariant under rotation, say ...
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1answer
45 views

Apparently a catastrophe: unital injective *-Hom implies factorization of Algebra into tensor product

In a recent lecture we saw the following result, which was hailed as somewhat of a catastrophe: We denote by $\operatorname{M}_m$ be the set of $m\times m$ matrices and by $(\hat A,\hat\varphi)$ a ...
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23 views

Definition of operator $x\otimes y$

Suppose that $\mathbb H$ is a separable Hilbert space. I am interested in the tensor $x\otimes y$, where $x,y\in\mathbb H$. From what I understand, the tensor $x\otimes y$ can be viewed as a bounded ...
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12 views

Anti-dual Space of Hilbert Space

This is probably a pretty soft question. Setup: Let $H$ be a Hilbert space over $\mathbb{C}$ with inner product $\langle x| y\rangle$ where $y\mapsto \langle x|y\rangle$ is linear and let $H'$ be ...
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1answer
19 views

Symmetrization is the unique $k$-tensor

$\newcommand{\Sym}[1]{\operatorname{Sym}{#1}}$ Let $V$ be a $n$-dim real vector space with dual space $V^*$. Let $\alpha$ be a covariant $k$-tensor, i.e., $\alpha \in T^k(V^*) \equiv (V^*)^{\otimes k}...
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1answer
45 views

How can we define tensor product of random variable?

Let $(\Omega ,\mathcal F,\mathbb P)$ and $(\tilde\Omega, \tilde{\mathcal F},\mathbb Q)$ two probability space. Let $X$ a random variable on $\Omega $ and $Y$ a random variable on $\tilde{\Omega }$. ...
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1answer
64 views

Why is $E \otimes_B B^n$ isomorphic to $E^n$ (for finite dimensional commutative $B$-algebras)?

$B$ field, $E$ field and finite dimensional (as a $B$-vector space) $B$-algebra. In categorical terms, why is $E \otimes_B B^n \cong E^n$? In $(B \downarrow \mathbf{CRing}) \cong B\text{-Alg}$, we ...
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1answer
11 views

A question about tensoring and retaining exactness

This is a question from this document on the Universal Coefficient Theorem. We have the following chain complex: We then tensor each module with $G$, and get the following complex: How come ...
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10 views

Express 2-Tensor in Basis of $R^2$

I think I may just be confused by the question... would appreciate help! I am given the function, $f(ae_1 + be_2 , ce_1 + de_2) = ac + bd$ (it doesn't say, but I assume we have $e_1, e_2$ the ...
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1answer
38 views

Proving matrix equality $(K_{n,n}\otimes I_n)a^{\otimes3}=a^{\otimes3}$

How to prove the matrix equality $(K_{n,n}\otimes I_n)a^{\otimes3}=a^{\otimes3}$? Here $K_{n,n}$ is a $n^2\times n^2$ commutation matrix, $I_n$ is a $n\times n$ identity matrix and $a$ is a $n\times1$...
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2answers
48 views

Prove that the map $f: \Bbb C \times \Bbb C \to \Bbb C \times \Bbb C$ by $f(z_1,z_2)=(z_1z_2,z_1\bar{z_2})$ is an $\Bbb R $ bilinear map

I am trying to solve the question 27 of Section 10.4 of Dummit and Foote but I am stuck in the first problem: let me state the question and then I will attach the picture of the page of the ...
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2answers
42 views

Prove that $2 \otimes 1 \neq 0$ in $2\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ the tensor is over $\Bbb Z$.

Prove that $2 \otimes 1 $ is zero in $\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ but not a zero in $2\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ the tensor is over $\Bbb Z$. It is easy to show the first part that $2 \...
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25 views

Show that $(A \otimes id)|\psi \rangle=(id \otimes A^T)|\psi\rangle$.

Let $\{u_i: 1\leq i \leq n\}$ be an orthonormal basis for $\mathbb{C}^n$ and let $$|\psi\rangle=\sum_{i=1}^{n}\frac{1}{\sqrt{n}}u_i \otimes u_i$$. Show that $(A \otimes id)|\psi \rangle=(id \otimes ...
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1answer
63 views

How to show that $\mathcal{F} \otimes \mathcal{F}^\vee \cong \mathcal{O}_X$

Let $\mathcal{F}$ be a rank $1$ locally free sheaf. If we define $\mathcal{F}^\vee = Hom_{\mathcal{O}_X}(F, \mathcal{O}_X)$, then how would one go about showing that $\mathcal{F} \otimes \mathcal{F}^\...
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1answer
32 views

Generalization of sum of outer product

Consider a matrix $A \in \mathbb{R}^{d \times m}$ such that $m \geq d$ and denote its columns i.e $A_{:, i}$ by $a_i$. Let $AA^T$ is invertible. Now, consider the sum $S(A) = \sum_{r=1}^m a_r a^T_r$...
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1answer
21 views

coefficient extension for fraction field $K(T) \otimes_K L$

let $L/K$ be an algebraic field extension. denote by $K(T)= Frac(K[T])$ the transcendental field extension of $K$. I would like to find out how to show that the equation $$K(T) \otimes_K L = L(T)$$ ...
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1answer
48 views

Tensor Product over Algebraically Closed Field

I have a question about a statement/formulation in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 122): We fix an integral proper normal curve $X$ over a field $k$. ...
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Tensor Product of Fields is a Field

I have two questions about a construction introduced in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 122): We fix an integral proper normal curve $X$ over a field ...
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15 views

how to use tensor products to determine the coefficient matrix of the line

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section2.7 Exercise 5, I was puzzled at solving it. Here is the problem description: With $A = \...
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2answers
47 views

The direct sum of two finitely generated algebras is finitely generated

Let $X$ and $Y$ be sets of variables. Let $R$ be a commutative ring. Suppose $S$ and $T$ are finitely generated $R$-algebras. Then $S\cong R[X]/I$ and $T\cong R[Y]/J$ for some $R[X]$-ideal $I$ and $R[...
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Differential bigraded algebra when it is a tensor product of d. graded algebras: Definition of differential

So I'm reading "A User's guide to spectral sequences", 2nd ed., of McCleary and there is defined in page 11 the notion of differential bigraded algebra, and then there is the example of the bigraded ...
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In what spaces are outer product of x with itself positive semidefinite?

In the case of vectors in $\mathbb R^n$, it is quite simple to see that for any vector $x$, $$ v^Txx^Tv = (v^Tx)^2 \geq 0$$ so clearly the form $xx^T$ must form a positive semidefinite matrix. But ...
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1answer
37 views

Direct sum of $\mathbb{Z}$ with the infinite direct sum of $\mathbb{Z}_n$

I'm reading this and on page 443, Example 2, they said that $\mathbb{Z}\oplus \mathbb{Z}_n\oplus \mathbb{Z}_n\oplus...\cong \mathbb{Z}[x]/(nx) $. I have three questions: (1) Is it true that $\mathbb{...
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Basic property of kronecker delta $( A \otimes B)(A^{-1} \otimes B^{-1})$

Given non singular matrices $A_{n \times n},B_{m \times m}$ $$ ( A \otimes B)(A^{-1} \otimes B^{-1}) = (AA^{-1}) \otimes (BB^{-1}) = I_n \otimes I_m = I_{(nm \times nm )} $$ I was just reading ...
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1answer
75 views

Question About Tensor Products over Different Base Rings

I can't prove the following remark which appears in Matsumura's 'Commutative Ring Theory': If $B$ is an $A$-algebra and $M$ and $N$ are $B$-modules, $M\otimes_{B}N$ is the quotient of $M\otimes_{A}...
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80 views

Equivalent definitions of separable extension of a field

Armand Borel in his textbook "Linear Algebraic Groups", pp. 4, states that: $F$ is said to be separable over $\boldsymbol{k}$ if it satisfies the following equivalent conditions ($p$ denotes the ...
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38 views

Homs and Tensor products, a specific isomorphism?

Let $L$ be a finite field extension of $\mathbb Q_p$ and $\mathfrak o$ its ring of integers. Let $M$ be an $\mathfrak o$-module and $N$ a $\mathbb Z_p$-module. We then have the map $$ \Phi \colon \...
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Interpretations for higher Tor functors [migrated]

Let's work in the category $R$-${\sf mod}$, for concreteness. I know that one can see the modules ${\rm Ext}^n_R(M,N)$ as modules of equivalence classes of $n$-extensions of $M$ by $N$ (Yoneda ...
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1answer
51 views

What is a linear isomorphism between $\underset{n\times m}{\times} \mathbb{R}$ and $\mathbb{R}^n\otimes\mathbb{R}^m$? [closed]

What is a linear isomorphism between $\underset{n\times m}{\times} \mathbb{R}$ and $\mathbb{R}^n\otimes\mathbb{R}^m$? Where $n\times m :=\{(i,j):0\le n-1,0\le j \le m-1\}$. Since $\underset{n\times ...
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28 views

Submodule Definition and Tensor Product

In Hungerford's Algebra, submodule is defined as follows: Let $R$ be a ring, $A$ an $R$-module and $B$ a nonempty subset of $A$. $B$ is a submodule of $A$ provided that $B$ is an additive ...
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1answer
79 views

Interpretation of $\nabla$ operator in an expression

Let a tensor (3x3) be of the form $U = \mathbf{u}\mathbf{v}$ ($\mathbf{u}$ and $\mathbf{v}$) being two fluid velocity vectors (of dimension 1x3). In my analysis, for such a tensor $U$, following ...
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1answer
46 views

The tensor product of two blocks of positive operators is positive

Let $$T = \begin{bmatrix} T_{11} & T_{12}\\ T_{21} & T_{22} \end{bmatrix},\quad S = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix}$$ be two positive operators on $E\...
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1answer
29 views

Locally constant functions and base ring extension

Let $X$ be any topological space, and let $R$ be any commutative ring with identity (in particular, a $\mathbb{Z}$-module). Let $C_c^{\infty}(X,R)$ be the $R$-module of locally constant functions with ...
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1answer
40 views

Is there a tensor product of $G$-sets?

We can take the tensor product of two vector spaces, and the tensor product of two modules. I'm wondering if the same can be done for group actions. Let $G$ be a group which acts on two sets $X$ and ...
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Tensor products $ f^∗(S\otimes T) = (f^∗S)\otimes (f^∗T) $ .

Let $V$,$W$ be finite dimensional vector spaces and let $f$ be a linear map, define $f^*$ to be the adjoint of $f$. If $S$,$T$ are tensors on $W$ show that $f^∗(S\otimes T) = (f^∗S)\otimes (f^∗T)$ ...
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2answers
50 views

Showing that $R^n \otimes_R R^m$ is isomorphic to $R^{nm}$ when $R$ is commutative

Let $R$ be a commutative ring. I want to show that $R^n \otimes_R R^m$ is isomorphic to $R^{nm}$ as $R$-module. So far, I've tried to define the only natural map that I can think of: $$ f : R^n \...
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1answer
40 views

Currying in a locally small category with coproducts

While studying for category theory course I stumbled upon the following question taken from a previous exam: Let $\mathcal{D}$ be a locally small category with all coproducts. Show that for every ...
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36 views

On a subset of the associated primes of tensor product of modules

For a module $M$ over a commutative ring $R$, let $\operatorname{Ass}_R (M):=\{\operatorname{ann}_R (m) \mid m\in M$ and $\operatorname{ann}_R(m) \in \operatorname{Spec}(R)\}$. If $M,N$ are ...
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0answers
17 views

Unconditional basis in the tensor product of Lp Banach spaces

Assume that X and Y are topological spaces with $\sigma$-finite measure, $L_p(X)$ is a Banach space of complex-valued functions so that $\int_X |f(x)|^p dx < \infty$, $1 \leqslant p < \infty$. ...
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1answer
123 views

Derivation of Vector Laplacian in Cylindrical Coordinates through Tensor Analysis

I'm currently trying to derive the Navier-Stokes equations in cylindrical coordinates through tensor analysis. I am only struggling with the last term on the right side, which is a vector Laplacian: $...
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26 views

Compute the size of the orbit under a finite group action

When $n = 3$, consider the three-dimensional tensors $X, X' \in \mathbb{F}_2^{2^n}$ given by $X = A \otimes e_1 + B \otimes e_2$ and $X' = B \otimes e_1 + (A+B) \otimes e_2$ where $A = \begin{...