What is Tensor algebra: Definition and 68 Discussions
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below).
The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure.
Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.
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...
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...
A brief explanation of vector and tensor concepts from A Student's Guide to Vectors and Tensors by Dan Fleisch. I found this when I was trying to better understand tensors and how they are used.
The trace of the sigma should be the same in both new and old basis. But I get a different one. Really appreciate for the help.
I’ll put the screen shot in the comment part
Hello there, recently I've been trying to demonstrate that, $$\textbf{W}^2 = -m^2\textbf{S}^2$$ in a rest frame, with ##W_{\mu}## defined as $$W_{\mu} = \dfrac{1}{2}\varepsilon_{\mu\alpha\beta\gamma}M^{\alpha\beta}p^{\gamma}$$ such that ##M^{\mu\nu}## is an operator of the form $$...
In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation)
##
\nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j}
##
I'm trying to prove that this covariant...
Let ##\varphi## be some scalar field. In "The Classical Theory of Fields" by Landau it is claimed that
$$
\frac{\partial\varphi}{\partial x_i} = g^{ik} \frac{\partial \varphi}{\partial x^k}
$$
I wanted to prove this identity. Using the chain rule
$$
\frac{\partial}{\partial x_{i}}=\frac{\partial...
According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is:
##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0##
Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...
The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...
I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...
I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it.
Tensors. As...
I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways:
##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...
Property (a) simply states that a second rank tensor that vanishes in one frame vanishes in all frames related by rotations.
I am supposed to prove: ##T_{i_1 i_2} - T_{i_2 i_1} = 0 \implies T_{i_1 i_2}' - T_{i_2 i_1}' = 0##
Here's my solution. Consider,
$$T_{i_1 i_2}' - T_{i_2 i_1}' = r_{i_1...
I'm looking for literature recommendations regarding tensor networks. I never came across singular value decomposition or spectral decomposition in my linear algebra classes, so I need to brush up on the relevant mathematical background as well.
To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation:
$$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$
and using the tetrad formalism to compute the coefficients of the...
I need the Ricci scalar for the FRW metric with a general lapse function ##N##:
$$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$
Could someone put this into Mathematica as I don't have it?
I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##.
##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}##
##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...
I have an equation $$
\chi_\nu\nabla_\mu\chi_\sigma+\chi_\sigma\nabla_\nu\chi_\mu+\chi_\mu\nabla_\sigma\chi_\nu=0
$$so we also have$$
g_{\nu\rho}g_{\mu\tau}g_{\sigma\lambda}\left(\chi^\rho\nabla^\tau\chi^\lambda+\chi^\lambda\nabla^\rho\chi^\tau+\chi^\tau\nabla^\lambda\chi^\rho\right)=0
$$Does...
I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things.
Based on my understanding, the contravariant component of a vector transforms as,
##A'^\mu = [L]^\mu~ _\nu A^\nu##
the covariant component of a vector...
Hello everyone!
It seems I can't solve this exercise and I don't know where I fail.
By inserting the metric on the lefthand side of I. and employing the chain rule, the equation eventually reads (confirmed by my notes from the tutorial):
$$m\frac{\mathrm{d}p_\delta}{\mathrm{d}t} =...
Hello, I am calculating the krauss operators to find the new density matrix after the interaction between environment and the qubit.
My question is: Is there an operational order between matrix multiplication and tensor product? Because apparently author is first applying I on |0> and X on |0>...
This was my attempt at a solution and was wondering where did I go wrong: -\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=-\frac{\partial}{\partial p_\mu}[\gamma^\nu p_\nu]^{-1}=\gamma^\nu\frac{\partial p_\nu}{\partial p_\mu}[\gamma^\sigma...
I can see that by the tensor transformation law of the Kronecker delta that
##\frac{\partial x^a}{\partial x^b}=\delta^a_b##
And thus coordinates must be independent of each other.
But is there a more straightforward and fundamental reason why we don’t consider dependent coordinates? Is it...
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.\...
Depending on the source, I'll often see EFE written as either covariantly:
$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8 \pi GT_{\mu\nu}$$
or contravariantly
$$R^{\alpha\beta} - \frac{1}{2}Rg^{\alpha\beta} = 8 \pi GT^{\alpha\beta}$$
Physically, historically, and/or pragmatically, is there a...
Homework Statement
Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$
where q is a constant vector.
Homework EquationsThe Attempt at a Solution
$$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}}...
What is the difference between ##{T{_{a}}^{b}}## and ##{T{^{a}}_{b}}## ? Both are (1,1) tensors that eat a vector and a dual to produce a scalar. I understand I could act on one with the metric to raise and lower indecies to arrive at the other but is there a geometric difference between the...
I need to prove that in a vacuum, the energy-momentum tensor is divergenceless, i.e.
$$ \partial_{\mu} T^{\mu \nu} = 0$$
where
$$ T^{\mu \nu} = \frac{1}{\mu_{0}}\Big[F^{\alpha \mu} F^{\nu}_{\alpha} - \frac{1}{4}\eta^{\mu \nu}F^{\alpha \beta}F_{\alpha \beta}\Big]$$ Here ##F_{\alpha...
Homework Statement
Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds:
## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ##
Homework Equations
## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ##
The...
Hi everyone, I am trying to self study some general relativity however I met some problem in the
contravarient and covarient basis.
In the lecture, or you can also find it on wiki page 'curvilinear coordinates',
the lecturer introduced the tangential vector ei =∂r/∂xi and the gradient vector ei...
Hi
I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...
Consider the expression
$$\left(T^{a}\partial_{\mu}\varphi^{a} + A_{\mu}^{a}\varphi^{b}[T^{a},T^{b}] + A_{\mu}^{a}\phi^{b}[T^{a},T^{b}]\right)^{2},$$
where ##T^{a}## are generators of the ##\textbf{su}(N)## Lie algebra, and ##\varphi^{a}##, ##\phi^{a}## and ##A_{\mu}^{a}## are numbers.
How...
The Lorentz transformation matrix may be written in index form as Λμ ν. The transpose may be written (ΛT)μ ν=Λν μ.
I want to apply this to convert the defining relation for a Lorentz transformation η=ΛTηΛ into index form. We have
ηρσ=(ΛT)ρ μημνΛν σ
The next step to obtain the correct...
Given the definition of the covariant basis (##Z_{i}##) as follows:
$$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$
Then, the derivative of the covariant basis is as follows:
$$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$
Which is also equal...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Example 10.1 in Section 10.3 ...Example 10.1 plus some preliminary definitions reads as...
If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as...
In the transformation of tensor components when changing the co-ordinate system, can someone explain the following:
Firstly, what is the point in re-writing the indicial form (on the left) as aikTklajl? Since we're representing the components in a matrix, and the transformation matrix is also...
I'm having trouble with the mathematics of tensor products as applied to Bell states.
Say I have the state
\begin{align*}
\left|\psi\right> &= \frac{1}{\sqrt{2}} \left(\left|0\right>_A \otimes \left|0\right>_B + \left|1\right>_A \otimes \left|1\right>_B\right)
\end{align*}
How would the...
Hello all!
I've just started to study general relativity and I'm a bit confused about dual basis vectors.
If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_j But in some pdf and...
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...
I am trying to derive the geodesic equation using variational principle.
My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$
Using the Euler-Lagrange equation, I have got this.
$$ \frac{d^2 x^u}{dt^2} + \Gamma^u_{mk} \frac{dx^m}{dt} \frac{dx^k}{dt} =...
Dear All,
I'm doing some tensor calculation on the divergence of gradient (of a vector) inverse. Am I allowed to first use the nabla operator on gradient and then inverse the whole product?
In other words, I'm searching for the divergence of a 2nd order tensor which is itself inverse of...
1. Problem statement:
Assume that u is a vector and A is a 2nd-order tensor. Derive a transformation rule for a 3rd order tensor Zijk such that the relation ui = ZijkAjk remains valid after a coordinate rotation.Homework Equations :
[/B]
Transformation rule for 3rd order tensors: Z'ijk =...