Tensor product Definition and 141 Threads

  1. jv07cs

    I Mapping tensor products into a Clifford algebra

    Considering a vector space ##W = V\oplus V^*## equipped with quadratic form Q such that we have a clifford algebra ##Cl(W, Q)##. How can I map elements of ##V\otimes V^*## into elements of ##Cl(W, Q)##? What about elements of ##V^* \otimes V##, ##V\otimes V## and ##V^* \otimes V^*## into ##Cl(W...
  2. cianfa72

    I MWI as applied to measurement of spin 1/2 entangled particle pair

    Hello, consider a pair of 1/2 spin entangled system of particles A and B given in the basis of eigenvectors of Pauli operator ##\sigma_z## as $$\ket{\psi} = \frac {1} {\sqrt (2)} \left ( \ket {+z} \otimes \ket {-z} - \ket {-z} \otimes \ket {+z} \right )$$ A measurement of particle A's spin along...
  3. cianfa72

    I Terminologies used to describe tensor product of vector spaces

    Hi, I'm in trouble with the different terminologies used for tensor product of two vectors. Namely a dyadic tensor product of vectors ##u, v \in V## is written as ##u \otimes v##. It is basically a bi-linear map defined on the cartesian product ##V^* \times V^* \rightarrow \mathbb R##. From a...
  4. jv07cs

    I Question Regarding Definition of Tensor Algebra

    I am currently reading this book on multilinear algebra ("Álgebra Linear e Multilinear" by Rodney Biezuner, I guess it only has a portuguese edition) and the book defines an Algebra as follows: It also defines the direct sum of two vector spaces, let's say V and W, as the cartesian product V x...
  5. A

    I Simplify tensor product statement

    Hi, if I wanted to show ##(Z \otimes Y)^{\dagger} = Z \otimes Y##, then I could simply multiply out the matrices belonging to the operators of quantum gates ##Z## and ##Y##. But my question is whether this is also solvable via the properties of the tensor product and the properties of the...
  6. ARoyC

    I Tensor Product of Two Hilbert Spaces

    How to prove that the tensor product of two same-dimensional Hilbert spaces is also a Hilbert space? I understand that I need to prove the Cauchy Completeness of the new Hilbert space. I am stuck in the middle.
  7. D

    I Understanding tensor product and direct sum

    Hi, I'm struggling with understanding the idea of tensor product and direct sum beyond the very basics. I know that direct sum of 2 vectors basically stacks one on top of another - I don't understand more than this . For tensor product I know that for a product of 2 matrices A and B the tensor...
  8. Vanilla Gorilla

    B Array Representation Of A General Tensor Question

    So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I, in the position of a complete beginner, am taking notes on it, and I just wanted to make sure I wasn't misinterpreting anything. At about 5:50, he states that "The array for Q is...
  9. Vanilla Gorilla

    B Transformation Rules For A General Tensor M

    So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I am a complete beginner and just want some clarification on if I'm truly understanding the material. Basically, is everything below this correct? In summary of the derivation of the...
  10. Euge

    POTW Semisimple Tensor Product of Fields

    Let ##L/k## be a field extension. Suppose ##F## is a finite separable extension of ##k##. Prove ##L\otimes_k F## is a semisimple algebra over ##k##.
  11. H

    B Tensor product of operators and ladder operators

    Hi Pfs i have 2 matrix representations of SU(2) . each of them uses a up> and down basis (d> and u> If i take their tensor product i will get 4*4 matrices with this basis: d>d>,d>u>,u>d>,u>u> these representation is the sum equal to the sum of the 0-representation , a singlet represertation with...
  12. J

    A Tensor product matrices order relation

    We mainly have to prove that this quantity## \bra{\varphi} A^{\otimes n } \ket{\varphi} \pm \bra{\varphi} B^{\otimes n } \ket{\varphi} ## is greater or equal than zero for all ##\ket{\varphi}##. Being ##\ket{\varphi}## a product state it is straightforward to demonstrate such inequality. I am...
  13. X

    I Is tensor product the same as dyadic product of two vectors?

    Is tensor product the same as dyadic product of two vectors? And dyadic multiplication is just matrix multiplication? You have a column vector on the left and a row vector on the right and you just multiply them and that's it? We just create a matrix out of two vectors so we encode two...
  14. G

    I The tensor product of tensors confusion

    > **Exercise.** Let T1and T2be tensors of type (r1 s1)and (r2 s2) respectively on a vector space V. Show that T1⊗ T2can be viewed as an (r1+r2 s1+s2)tensor, so that the > tensor product of two tensors is again a tensor, justifying the > nomenclature... What I’m reading:《An introduction to...
  15. L

    A Tensor product in Cartesian coordinates

    I am confused. Why sometimes perturbation ##V'=\alpha xy## we can write as ##V'=\alpha x \otimes y##. I am confused because ##\otimes## is a tensor product and ##x## and ##y## are not matrices in coordinate representation. Can someone explain this?
  16. Decimal

    I Completeness relations in a tensor product Hilbert space

    Hello, Throughout my undergrad I have gotten maybe too comfortable with using Dirac notation without much second thought, and I am feeling that now in grad school I am seeing some holes in my knowledge. The specific context where I am encountering this issue currently is in scattering theory...
  17. K

    I Prove that dim(V⊗W)=(dim V)(dim W)

    This proof was in my book. Tensor product definition according to my book: $$V⊗W=\{f: V^*\times W^*\rightarrow k | \textrm {f is bilinear}\}$$ wher ##V^*## and ##W^*## are the dual spaces for V and W respectively. I don't understand the step where they say ##(e_i⊗f_j)(φ,ψ) = φ(e_i)ψ(f_j)##...
  18. K

    Show that a (1,2)-tensor is a linear function

    I know that a tensor can be seen as a linear function. I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.
  19. phun_physics

    I Representing Quantum Gates in Tensor Product Space

    Where do I start. I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example: Both qubits, ##q_0## and ##q_1## start in the ground state, ##|0 \rangle =\begin{pmatrix}1 \\ 0 \end{pmatrix}##. Then we...
  20. V

    A Equivalence Relation to define the tensor product of Hilbert spaces

    I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces: ##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}## The definition for the equivalence relation is given in the lecture vidoe as ##(\sum_{j=1}^{J}c_j\psi_j...
  21. hilbert2

    I Notation for vectors in tensor product space

    Suppose I have a system of two (possibly interacting) spins of 1/2. Then the state of each separate spin can be written as a ##\mathbb{C}^2## vector, and the spin operators are made from Pauli matrices, for instance the matrices ##\sigma_z \otimes \hat{1}## and ##\hat{1} \otimes \sigma_z##...
  22. W

    How Do You Express the Tensor Product of Hamiltonians?

    ##U_1 \otimes U_2 = (1- i H_1 \ dt) \otimes (1- i H_2 \ dt)## We can write ## | \phi_i(t) > \ = U_i(t) | \phi_i(0)>## where i can be 1 or 2 depending on the subsystem. The ## U ##'s are unitary time evolution operators. Writing as tensor product we get ## |\phi_1 \phi_2> = (1- i H_1 \ dt) |...
  23. A

    I No problem, happy to help! Good luck with your studies.

    Hi, I'm currently working through a tensor product example for a two qubit system. For the expression: $$ \rho_A = \sum_{J=0}^{1}\langle J | \Psi \rangle \langle \Psi | J \rangle $$ Which has been defined as from going to a global state to a local state. Here $$ |\Psi \rangle = |\Psi^+...
  24. M

    A Defining the Tensor Product of Gradients for Different Coordinate Systems

    Does anyone know where I can find the definition of ##\nabla \otimes \nabla f##? I tried googling this but nothing comes up. I know it will change depending on the coordinate system, so does anyone know the general definition OR a table for rectangular, spherical, cylindrical coordinates...
  25. George Keeling

    A Question about covariant derivatives

    I am reading I am reading Spacetime and Geometry : An Introduction to General Relativity -- by Sean M Carroll and have arrived at chapter 3 where he introduces the covariant derivative ##{\mathrm{\nabla }}_{\mu }##. He makes demands on this which are \begin{align} \mathrm{1.\...
  26. K

    A Definition of Tensor and.... Cotensor?

    Why are there (at least) two definitions of a tensor? For some people a tensor is a product of vectors and covectors, but for others it's a functional. While it's true that the two points of view are equivalent (there's an isomorphism) I find having to switch between them confusing, as a...
  27. Prez Cannady

    I Confused by this result for the tensor product of two vectors

    Given two probability distributions ##p \in R^{m}_{+}## and ##q \in R^{n}_{+}## (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as $$p \otimes q := \begin{pmatrix} p(1)q(1) \\ p(1)q(2) \\ \vdots \\ p(1)q(n) \\ \vdots \\...
  28. N

    B Tensor Product, Basis Vectors and Tensor Components

    I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space. 1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ) 2. Tμνσρ = T(θμ,θν,eσ,eρ) My attempt is as follows: 2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)...
  29. S

    I Understanding Kunneth Formula and Tensor Product in r-Forms

    Hello! Kunneth fromula states that for 3 manifolds such that ##M=M_1 \times M_2## we have ##H^r(M)=\oplus_{p+q=r}[H^p(M_1)\otimes H^q(M_2)]##. Can someone explain to me how does the tensor product acts here? I am a bit confused of the fact that we work with r-forms, which are by construction...
  30. F

    Insights Is There Meaning Behind (0,1) as a Tensor?

    fresh_42 submitted a new PF Insights post What Is a Tensor? Continue reading the Original PF Insights Post.
  31. L

    Partial traces of density operators in the tensor product

    Homework Statement Consider a system formed by particles (1) and (2) of same mass which do not interact among themselves and that are placed in a potential of infinite well type with width a. Let H(1) and H(2) be the individual hamiltonians and denote |\varphi_n(1)\rangle and...
  32. arpon

    I Is Second rank tensor always tensor product of two vectors?

    Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions: 1. Are those two vectors ##A_i## and ##B_j## unique? 2. How to find out ##A_i## and ##B_j## 3. As ##A_i## and...
  33. D

    A Difference Between Outer and Tensor

    Say, we have two Hilbert spaces ##U## and ##V## and their duals ##U^*, V^*##. Then, we say, ##u\otimes v~ \epsilon~ U\otimes V##, where ##'\otimes'## is defined as the tensor product of the two spaces, ##U\times V \rightarrow U\otimes V##. In Dirac's Bra-Ket notation, this is written as...
  34. T

    A Hamiltonian with a tensor product - a few basic questions

    I am given a hamiltonian for a two electron system $$\hat H_2 = \hat H_1 \otimes \mathbb {I} + \mathbb {I} \otimes \hat H_1$$ and I already know ##\hat H_1## which is my single electron Hamiltonian. Now I am applying this to my two electron system. I know very little about the tensor product...
  35. F

    I What is the concept of tensor product and how is it used in mathematics?

    Hello, I have encountered the concept of tensor product between two or more different vector spaces. I would like to get a more intuitive sense of what the final product is. Say we have two vector spaces ##V_1## of dimension 2 and ##V_2## of dimension 3. Each vector space has a basis that we...
  36. V

    I (2,0) tensor is not a tensor product of two vectors?

    Hi. I'm trying to understand tensors and I've come across this problem: "Show that, in general, a (2, 0) tensor can't be written as a tensor product of two vectors". Well, prior to that sentence, I would have thought it could... Why not?
  37. nomadreid

    I Tensor product and ultraproduct construction

    I do not know if this is the proper rubric to ask this question, but I picked the one that seemed the most relevant. I have noticed some superficial resemblance between the tensor product and the ultraproduct definitions. Does this resemblance go any further? While I am on the subject of...
  38. G

    I Why the tensor product (historical question)?

    Hi. Why did the founding fathers of QM know that the Hilbert space of a composite system is the tensor product of the component Hilbert spaces and not a direct product, where no entanglement would emerge? I mean today we can verify entanglement experimentally, but this became technologically...
  39. T

    A Question about properites of tensor product

    They are being 2 by 2 matrices and I being the identity. Physically they are Pauli matrices. 1. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$ = $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$? I...
  40. J

    I Taking the Tensor Product of Vectors

    What is meant by taking the tensor product of vectors? Taking the tensor product of two tensors is straightforward, but I am currently reading a book where the author is talking about tensor product on tensors then in the next paragraph declares that tensors can then be constructed by taking...
  41. F

    I What is the outer product of a tensor product of vectors?

    If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...
  42. pellman

    I Tensor Product in QM: 1D vs 3D Hilbert Spaces

    A particle in a 1-D Hilbert space would have position basis states ## |x \rangle ## where ## \langle x' | x \rangle = \delta(x'-x) ## A 3-D Hilbert space for one particle might have a basis ## | x,y,z \rangle ## where ##\langle x', y', z' | x,y,z \rangle = \delta(x'-x) \delta (y-y') \delta(z-z')...
  43. Math Amateur

    I Basis of a Tensor Product - Theorem 10.2 - Another Question

    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 another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as...
  44. Math Amateur

    I Basis of a Tensor Product - Cooperstein - Theorem 10.2

    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 an aspect of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as follows: I do not...
  45. Math Amateur

    MHB Basis of a Tensor Product - Theorem 10.2 - Another Question .... ....

    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 another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as...
  46. Math Amateur

    MHB Basis of a Tensor Product - Cooperstein - Theorem 10.2

    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 an aspect of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as follows:I do not...
  47. Math Amateur

    I Proof of Existence of Tensor Product .... Further Question ...

    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 another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...The relevant part of...
  48. Math Amateur

    MHB Proof of Existence of Tensor Product: Cooperstein Theorem 10.1

    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 another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...
  49. Math Amateur

    I Tensor Product - Knapp - Theorem 6.10 .... Further Question

    I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with a further aspect of the proof of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: The above proof mentions Figure...
  50. Math Amateur

    MHB Theorem 6.10 in Knapp's Basic Algebra: Exploring Bilinearity & Descending Maps

    I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ... I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ... The text of Theorem 6.10 reads as follows: The above proof mentions Figure 6.1 which is...
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