Tensor product Definition and 133 Threads

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ...Theorem 10.1 reads as follows: In the above text...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ... Theorem 10.1 reads as follows:In the above...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Lemma 10.1 on the uniqueness of a tensor product ... ... Before proving the uniqueness (up to an...

I am trying to show that if (C^ab)(A_a)(B_b) is a scalar for arbitrary vectors A_a and B_b then C^ab is a tensor. I want to take the product of the two vectors then use the quotient rule to show that C^ab must then be a tensor. This lead to the question of whether or a not the product of two...

At the risk of sounding ignorant I'd like to propose a question to someone well versed in Homological Algebra and General Relativity. I'm starting to study the tensor product functor in the context of category theory because I'm interested in possibly doing a paper on TQFT for a directed...

On pages 67 & 68 of Hassani's mathematical physics book, he gives the following definition: "Let ## \mathcal{A} ## and ## \mathcal{B} ## be algebras. The the vector space tensor product ## \mathcal{A} \otimes \mathcal{B} ## becomes an algebra tensor product if we define the product ##...

Hey it might be a stupid question but I saw that the tensor product of 2 vectors with dim m and n gives another vector with dimension mn and in another context I saw that the tensor product of vector gives a metrix. For example from sean carroll's book: "If T is a (k,l) tensor and S is a (m, n)...

Hi, I am working through a textbook on general relativity and have come across the statement: "A general (2 0) tensor K, in n dimensions, cannot be written as a direct product of two vectors, A and B, but can be expressed as a sum of many direct products." Can someone explain to me how this...

Just a question : do we have in Dirac notation $$\langle u|A|u\rangle\langle u|B|u\rangle=\langle u|\langle u|A\otimes B|u\rangle |u\rangle$$ ?

I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space $V\otimes W$, along with a multilinear map $\pi:V\times W\to V\otimes W$ such that whenever there is...

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...

I'm trying to re-derive a result in a paper that I'm struggling with. Here is the problem: I wish to calculate (\nabla \otimes \nabla) h where \nabla is defined as \nabla = \frac{\partial}{\partial r} \hat{\mathbf{r}}+ \frac{1}{r} \frac{\partial}{\partial \psi} \hat{\boldsymbol{\psi}} and...

Homework Statement Consider ##T = \delta \otimes \gamma## where ##\delta## is the ##(1,1)## Kronecker delta tensor and ##\gamma \in T_p^*(M)##. Evaluate all possible contractions of ##T##. Homework Equations Tensor productThe Attempt at a Solution ##\gamma## is therefore a ##(0,1)## tensor...

I am working through an explanation in Nielson and Chuang's Quantum Computation book where they apply a CNOT gate to a state α|0>|00> + β|1>|00>. (The notation here is |0> = the column vector (1,0) and |1>=(0,1), while |00> = |0>|0>, and |a>|b>=|a>⊗|b>, ⊗ being the tensor (outer) product. I am...

Homework Statement I have the operators ##D_{\beta}:V_{\beta}\rightarrow V_{\beta}## ##R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}## ##R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}## where each...

I have recently delved into linear algebra and multi-linear algebra. I came to learn about the concepts of linear and bi-linear maps along with bases and changes of basis, linear independence, what a subspace is and more. I then decided to move on to tensor products, when I ran into a problem...

Hi,let: 0->A-> B -> 0 ; A,B Z-modules, be a short exact sequence. It follows A is isomorphic with B. . We have that tensor product is right-exact , so that, for a ring R: 0-> A(x)R-> B(x)R ->0 is also exact. STILL: are A(x)R , B(x)R isomorphic? I suspect no, if R has torsion. Anyone...

suppose we consider the measurement operator A=diag(1,-1). Then the tensor product of A by itself is in components : A\otimes A=a_{ij}a_{kl}=c_{ijkl} giving c_{1111}=c_{2222}=1, c_{1122}=c_{2211}=-1 and all other component 0. to diagonalize a tensor of order 4, we write ...

Hi everyone, :) Xristos Lymperopoulos on Facebook writes (>>link<<);

First, thanks to both Deveno and ThePerfectHacker for helping me to gain a basic understanding of tensor products of modules. In a chat room discussion ThePerfectHacker suggested I show that $$ {\mathbb{Z}}_a \otimes_\mathbb{Z} {\mathbb{Z}}_b $$ where a and b are relatively prime integers -...

In QM the tensor product of two independent electron's spin state vectors represents the product state which represents the possible unentangled states of the pair. I don't understand why the tensor product produces that result. |A⟩=|a⟩⊗|b⟩

Hello, It may be trivial to many of you, but I am struggling with the following integral involving two spheres i and j separated by a distance mod |rij| ∫ ui (ρ).[Tj (ρ+rij) . nj] d2ρ The integration is over sphere j. ui is a vector (actually velocity of the fluid around i th...

Hi, I have been learning about tensor products from Dummit and Foote's Abstract Algebra and I'm a little confused. I understand the construction of going to the larger free group and "modding out" by the relations that will eventually end up giving us module structure. But just in the...

Hello everyone, Say I have two irreducible representations \rho and \pi of a group G on vector spaces V and W. Then I construct a tensor product representation \rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right) by \left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v...

Hi everyone, I don't quite understand how tensor products of Hilbert spaces are formed. What I get so far is that from two Hilbert spaces \mathscr{H}_1 and \mathscr{H}_2 a tensor product H_1 \otimes H_2 is formed by considering the Hilbert spaces as just vector spaces H_1 and H_2...

Hi everyone, I'm reading through tensor product spaces and one question really bogs me. Why is it that the total Hilbert space of a system composed of two independent subsystems is the tensor product of the Hilbert spaces of the subsystems? It is always posed, but I've never seen a proof...

In many physics literature I have encountered, one of the properties of Levi-Civita tensor is that ε_{ijk}ε_{lmn}is equivalent to a determinant of Kronecker symbols. However this is only taken as a given theorem and is never proved. Is there any source which has proven this property?

I am trying to prove that C\otimesC (taken over R) is equal to C^2. The method I have seen is to show the following equivalences: C\otimesC = C\otimes(R[T]/<T^2+1>) = C[T]/<T^2+1> = C. (All tensor products taken over R). The only part I am having trouble with is showing that...

I'm trying to understand tensor product of vector spaces and how it is done,but looks like nothing that I read,helps! I need to know how can we achieve the tensor product of two vector spaces without getting specific by e.g. assuming finite dimensions or any specific underlying field. Another...

Homework Statement Show that \mathbb{Z}_{10}\otimes_{\mathbb{Z}}\mathbb{Z}_{12} \cong \mathbb{Z}_{2} The Attempt at a Solution Clearly, for any 0\neq m\in\mathbb{Z}_{10} and 0\neq n \in \mathbb{Z}_{12} we have that m\otimes n = mn(1\otimes 1), and if either m=0 or n=0 we have that m\otimes n =...

So I've been reading Cohen-Tannoudji's "Quantum mechanics vol 1" and have understood the part that proves that the hilbert space of a 3-dimensional particle can be described/decomposed as a tensor product of hilbert spaces using position vectors (or analogously momentum vectors) in the x, y and...

I'm reading about tensor product of modules, there's a theorem in the book that leaves parts of the proof to the reader. I've attached the file, I didn't put this in HW section because first of all I thought this question was more advanced to be posted in there and also because I want to discuss...

I am working on all of the problems from Georgi's book in Lie algebras in particle physics (independent study), but I am stuck on one of them. The question is the following: "Find (2,1)x(2,1) (in su(3) using Young tableaux). Can you determine which representations appear antisymmetrically in...

I have to demonstrate that if A^{rs} is an antisymmetric tensor, and B_{rs} is a symmetric tensor, then the product: A^{rs}B_{rs}=0 So I called the product: C^{rs}_{rs}=A^{rs}B_{rs}=-A^{sr}B_{sr}=-C^{rs}_{rs} In the las stem I've changed the indexes, because it doesn't matters which is which...

I never thought about this stuff much before, but I am getting confused by a couple of things. For example, would the state ∣E,l,m⟩ be the tensor product of ∣E⟩, ∣l⟩, ∣m⟩, ie. ∣E⟩∣l⟩∣m⟩? I always just looked at this as a way to keep track of operators that had simultaneous eigenfunctions in a...

Hi all, it is of course true that every linear map between two vector spaces can be expanded by means of the tensor product. For instance, the metric in General Relativity (mapping covectors to vectors) can be expanded as g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}. However, does this...

Are photons in the laser output beam mutually independent and can the multi-photon output be described as the tensor product of individual photon states?

Homework Statement I don't need help with the main problem, just a calculation: I need to expand out elements of the form (x_i \otimes 1)(1 \otimes x_j), etc. Homework Equations The Attempt at a Solution Is there a property of the tensor product that I can use to expand out products...

Hello. I keep on encountering the need to find the Tensor or Kronecker product of two vectors. Based on the definition, If I found the product of two 2D vectors, I would get a 4-dimensional vector. Some authors claim this is the correct interpretation. However the dyadic product, which many...

Hi all, I was reading the book by Herbert Federer on Geometric Measure Theory and it seems he proves the existence of the Tensor Product quite differently from the rest. However it is not clear to me how to prove the existence of the linear map "g" in his construction. He defines F as the...

Homework Statement Hi, I attached a file involving my problem and attempt at a solution.Thanks. Homework Equations The Attempt at a Solution

Tell me if this is true: We are given vector spaces V1, V2, ..., Vn of dimensions d1, d2, ..., dn respectively. Let V = V1 \otimes V2 \otimes ... \otimes Vn Claim: Any element v \in V can be represented in the following form: \sumi=1...R (v1,i \otimes ... \otimes vn,i) Where R =...

It is a well fact that tensor product is associative up to isomorphism, but how should I use Universal property(you know, diagrams that commute) to show that it is true?

When I was first introduced to the tensor product, I was actually introduced to a special case: the tensor product of vector spaces over \mathbb{C}, which was explained to be as the space of multilinear maps on the cross product of the dual spaces, for example. At the time I wasn't aware this...

Just as a concrete example, say A and A' are two 2x2 matricies from R^2 to R^2, A = \left [ \begin{array}{cc} a \,\, b \\ c \,\, d \end{array} \right ] A' = \left [ \begin{array}{cc} x \,\, y \\ z \,\, w \end{array} \right ] What would A \otimes_\mathbb{R} A' look like (say wrt the standard...

Suppose f_1 is a linear map between vector spaces V_1 and U_1, and f_2 is a linear map between vector spaces V_2 and U_2 (all vector spaces over F). Then f_1 \otimes f_2 is a linear transformation from V_1 \otimes_F V_2 to U_1 \otimes_F U_2. Is there any "nice" way that we can write the kernel...

Let U and V be vector spaces over the complex numbers C. Then the tensor product over C, U\otimes_CV is also a complex vector space. Note that U, V, and U\otimes_CV can be regarded as vector spaces over the real numbers R as well. Also note that we can form U\otimes_RV. Question: are U\otimes_CV...

Given R a ring and M,N two R-modules, we may form their tensor product over Z or R. They can be defined as the group presentations < A x B | (a + a',b)=(a,b) + (a',b), (a,b+b')=(a,b) + (a,b') >, < A x B | (a + a',b)=(a,b) + (a',b), (a,b+b')=(a,b) + (a,b'), (ra,b)=(a,rb) > respectively and the...