Tensor product Definition and 141 Threads

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...

The Wikipedia article Metric tensor (general relativity) has the following equation for the metric tensor in an arbitrary chart, g = g_{\mu\nu} \, \mathrm{d}x^\mu \otimes \mathrm{d}x^\nu It then says, "If we define the symmetric tensor product by juxtaposition, we can write the metric in...

So I'm reading about tensor products and wanting to make sure I understand the notion completely. I understand that V^* \otimes W is the space of linear functions from V \text{to} W. And since V^{**} \backsimeq V, we have that V \otimes W is the space of linear functions from V^* \text{to}...

Hi, I am reading this article for homework about a ring in a megnetic field. It starts off by giving a hamiltonian (an adiabatic part -never mind) H_{0}= \frac{1}{2M} [ \Pi -A]^{2} -\mu B( \phi) \cdot \sigma A- is a known operator where \Pi=\frac{1}{2a} \frac{d}{d \phi}...

Hello, I'm studying the Heisenberg Model. Given the Hamiltonian H = - 2 \frac{J}{\hbar^2} \vec{S}_1 \vec{S}_2 with \begin{equation} \vec{S} = \frac{\hbar}{2} \; \left( \begin{array}{ccc} \sigma_x \\ \sigma_y \\ \sigma_z \end{array} \right) \end{equation} \sigma_{x,y,z} \quad...

I was wondering about useful norms on tensor products of finite dimensional vector spaces. Let V,W be two such vector spaces with bases \{v_1,\ldots,v_{d_1}\} and \{w_1,\ldots,w_{d_2}\}. We further assume that each is equipped with a norm, ||\cdot||_V and ||\cdot||_W. Then the tensor product...

Tensor product of vector space problms Homework Statement I'm currently reading Halmos's book "Finite dimensional vector spaces" and I find it excellent. However, I'm having some problems with his definition of the tensor product of two vector spaces, and I hope you could help me clear it...

Hello, I'm currently reading Halmos's book "Finite dimensional vector spaces" and I find it excellent. However, I'm having some problems with his definition of the tensor product of two vector spaces, and I hope you could help me clear it out. Here's what he writes: "Definition: The tensor...

I think the answer depends on the metric signature (right?), so for this topic, let's define the signature of the metric to be positive I'm curious what properties a tensor T must have, for the following to be true: T^{ab} T_{ab} \ge 0 Note: I am not talking about the stress energy tensor. I'm...

So this is supposed be an introductory problem for tensor products that I was trying to do to verify I am understanding tensor products...turns out I'm not so much Show that M_n(K) is isomorphic as an F-algebra to K \otimes_F M_n(F) where F is a field and K is an extension field of F and...

Having a little trouble deriving a result in a book. If I have an operator of the form e^{\alpha A \otimes I_n} Where alpha is a complex constant, A a square hermitian matrix and I the identity matrix. Now if I want to operator that on a tensor product, say for instance c_{n,1} |1...

Dear all, I've read the math that defines a tensor product by means of the universal property and I've studied the tensor product construction through a quotient of the free vector space on the cartesian product of two vector spaces. All other constructions of the tensor products are naturally...

Dear all, why is it that the tensor product is used to describe two quantum system described by Hilbert spaces H1 and H2? What were the example systems or situations that were generalized and that led to this postulate? Thanks. Goldbeetle

In the first equation on this page, https://www.physicsforums.com/library.php?do=view_item&itemid=335 is there a loss of generality when there exists a metric tensor, since in that case V \otimes V \otimes V^* \neq V \otimes V^* \otimes V, because T^{ij}\;_{k} \neq T^{i}\;_{k}\;^{j}.

I still don't fully understand the explicit construction of the tensor product space of two vector spaces, in spite of the efforts by several competent posters in another thread about 1.5 years ago. I'm hoping someone can provide the missing pieces. First, a summary of the things I think do...

Homework Statement Suppose that [\sigma_a]_{ij} and [\eta_a]_{xy} are Pauli matrices in two different two dimensional spaces. In the four dimensional tensor product space, define the basis: |1\rangle=|i=1\rangle|x=1\rangle |2\rangle=|i=1\rangle|x=2\rangle |3\rangle=|i=2\rangle|x=1\rangle...

For some reason, tensors seem to be a terribly mysterious topic, mentioned all the time, but rarely explained in clear terms. Whenever I read a paper which uses them, I get the feeling I'm listening to a blind man talk about an elephant. They have to do with multilinear maps. They are a...

When considering finite dimensional vector spaces V and W over a field K, there exists a natural isomorphism between their tensor product and the space of bilinear maps from the cartesian product of the dual spaces to the underlying field. However, the text I'm reading asserts that if V and W...

If V1 and V2 are both subspaces of a vector space V, then in order for their tensor product to be defined, does the intersection of V1 and V2 have to be 0?

Hi PF bloggers, I'm trying to decompose a representation of GL(2,C) on C^2\otimes Sym^{N-2}(C^2) into IRREPS and I'm wondering if there's anything similar to Clebsh-Gordan coefficients which could assist one in this task? Any good references one could point out? Happy holidays! P.S...

Hello, So I'm trying to understand the construction of the tensor product of 2 vector spaces as stated in the http://en.wikipedia.org/wiki/Tensor_product" . Now, in the article it states that the tensor product of two vector spaces V and W is the quotient space F( VxW )/R (F( VxW ) being the...

Homework Statement Prove that V* \otimes W is isomorphic to Hom(V,W) in the case that one of V and W is finite-dimensional. The Attempt at a Solution A pair (l,w) in V*xW defines a map V->W, v->l(v)w. This map is bilinear. Because it's bilinear, it defines a bilinear map V* \otimes W ->...

Hi, can anyone please explain me how to understand this term? I tried to expand it, but seems I may not be right, so can anyone help me with expasion of this rhs term below? T is suppsoed to be symmetric, but when I expand it it doesn't seem to be symmetric, please help. consider 2 mutually...

Hey guys, How exactly do you take the trace of a tensor product? Do I take the trace of each tensor individually and multiply their traces? For example, how would I take the trace of this tensor product: -B^{c}_b B_{ac}

There's a question that asks me to show that there exists a unique linear transformation from: f\otimes g: V_1\otimes W_1\rightarrow V_2\otimes W_2 where f and g are linear transformations f:V1->V2, g:W1->W2 that satisfies: (f\otimes g)(u\otimes v)=f(u)\otimes g(v) well I think that what I...

Hello all. Most modern treatments of the tensor product use equivalence classes to define a quotient space in order to define the tensor product. However in Tensor Analysis on Manifolds, Bishop and Goldberg are much less complicated. I have attached a near word for word copy of their...

I'm reading the Wikipedia article, trying to understand the definition of the tensor product V\otimes W of two vector spaces V and W. The first step is to take the cartesian product V\times W. The next step is to define the "free vector space" F(V\times W) as the set of all linear combinations...

Hello, I have an exercise where we have to pullback a metric g^{ij} \, \mathrm dx_i \, \mathrm dx_j under a function f: M \rightarrow N (actually in this case M = \mathbf{R}^2, N = \mathbf{R}^3, but that's not really relevant). I managed to do it, provided that the pullback commutes with the...

what is the tensor product's physical significance? I know what it does mathematically, but what does it mean. I have looked on textbooks and wikipedia but i still can't understand the physical signifcance.

Could someone explain to me what this is and explain the formula to me? I don't think I understand the formula. I don't think I quite understand why that's the antisymmetrized tensor product. Maybe its because i don't want o think about it too much.

How can we prove that the tensor product between two tensors of lower rank forms the basis for ANY tensor of higher order? also WHY is it it true? ANY TENSOR of higher order.

could someone please explain to me what the tensor product is and why we invented it? most resources just state it without listing a motivation.

Could someone tell me what the tensor product is and give an example?

Can someone please explain to me what is the tensor product and any good elementary tensor algerbra books?

Hi. I am trying to perform a tensor product between two 2x2 matrices using Mathematica. When i simply use the symbol for tensor product and put it between the two matrices, the program just reproduce the same expression when i execute it. I tried to multiply the individual elements of...

on the "Tensor Product" In response to some remarks made in the thread "How do particles become entangled?", as well as a number of private messages I have received, I feel there is some need to post some information on the notion of a "tensor product". Below, a rather intuitive look at the...

It's possible to do the tensor product of two contravariant vectors? It's possible to do the tensor product of two covariant vectors?