Use this tag for questions that involve tensors. Tensors are fundamental tools for linear computations, generalizing vectors and matrices to higher ranks. Mathematica 9 introduces powerful methods to algebraically manipulate tensors with any rank and symmetry.

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40
votes
2answers
2k views

Internal`Bag inside Compile

Since Internal`Bag, Internal`StuffBag and Internal`BagPart can be compiled down, it is a ...
16
votes
4answers
5k views

How to calculate scalar curvature Ricci tensor and Christoffel symbols in Mathematica?

I am seeking a convenient and effective way to calculate such geometric quantities. I've used packages like TensoriaCalc, but they don't work at all time. ...
14
votes
2answers
2k views

How to represent and manipulate abstract indexed vector (or tensor) expressions?

I have a couple abstract indexed quantities, both differential elements $dx = dx^\mu e_\mu + x^\mu de_\mu$ $du = du^\mu e_\mu + u^\mu de_\mu$ I can compute the expression $(dx + du) \cdot (dx + du) ...
13
votes
2answers
765 views

Higher order SVD

Does anyone know how to do a higher order SVD in Mathematica ? A good reference seems to be here http://www.sandia.gov/~tgkolda/pubs/pubfiles/SAND2007-6702.pdf but I don't understand their formalism ...
10
votes
3answers
306 views

Looking for an elegant way to construct this tensor-product-ish list

I would like to make the following matrix: ...
10
votes
1answer
593 views

Nesting Parallel processes

I just attempted to run code that had nested ParallelMap[] functions. It generates the error message: ParallelMap::subpar: Parallel computations cannot be nested; proceeding with sequential ...
9
votes
1answer
251 views

Verifying and deriving basic (block) matrix identities

How can I use the new symbolic matrix/tensor capabilities to verify matrix identities, such as (1) or (2) Even better, how can I ask Mathematica to derive expressions for X, Y, Z, and U like ...
9
votes
2answers
212 views

How to take derivative of parameterized coordinate?

Suppose I have a vector in $\mathbb{R}^n$ but $n$ is not known in advance. I want to be able to write functions which operate on the components of that vector, and then I'd like to be able to take ...
8
votes
3answers
268 views

Reduce the output from tuples by including symmetry?

I need all the possible 3x3 binary tensors, but I'd like to have this account for symmetries. I've started by using the Tuples command. Tuples[{1, 0}, {3, 3}] ...
8
votes
1answer
186 views

inverse of abstract matrix

If you assume the matrix $A$ is invertible, then $A^{-1} \cdot A = I$. Is there an assumption for invertibility in Mathematica 9? How can one make the following evaluate to the identity matrix $I_3$? ...
7
votes
1answer
1k views

Tensor analysis

Has anybody used tensors in Mathematica? How to properly work with them? I find Mathematica not very friendly in this field, as I am defining my own functions for lowering & raising indices, ...
7
votes
1answer
250 views

Compiling Map over expression that yields a ragged array

I'm trying to speed up a function that looks in the neighborhood of each 3D point in a large dataset and finds all the points within 1 unit in each direction, x, y, z. I've started by using ...
6
votes
3answers
813 views

Ways to compute inner products of tensors

One way to evaluate the following sums is combining Table and Sum: $u_{abcd} = \sum_{e=1}^3 v_{aeb}w_{ced}$ $q_{ab} = \sum_{d,e=1}^3 v_{d e a}w_{deb}$ It will look like ...
6
votes
2answers
185 views

How to get rid of nested matrices

If I type into Mathematica TensorProduct[IdentityMatrix[2],IdentityMatrix[2]] It gives me a result that has nested matrices. How do I turn that into a normal ...
6
votes
1answer
178 views

Computations in the exterior algebra

I want to be able to compute with explicit exterior algebras of vector spaces. For example, given a real vector space $V$ of $3 \times 3$ matrices, I want to consider expressions of the form $v\wedge ...
6
votes
1answer
489 views

TensorContract of inverse matrix

Matrix inverse in mathematica If $A$ is an invertible $n \times n$ matrix, then $A\cdot A^{-1} = I$. To get this statement in Mathematica, you need the assumption ...
6
votes
1answer
332 views

Solving antisymmetric tensorial equation

Assume we have the following Tensor objects: \begin{equation} F_{i}{}^{j}\;and\;S_{ij}{}^{k}, \end{equation} where the components of $F$ are known, and we would like to solve for the components of $S$ ...
5
votes
3answers
474 views

Multidimensional array reduction through summation over one of its dimensions

1. Introduction I am using an array of dimension 3 (might become more) to store some values. I would like to implement a function that takes as argument the array and a couple of numbers smaller than ...
5
votes
5answers
392 views

Elementwise join

I have two tensors of arbitrary but equal rank n (and equal dimensions): A and B, and I want to get a third tensor of rank n + ...
5
votes
1answer
221 views

Symbolic linear algebra

I would like to know how I can ask Mathematica to expand (and simplify) such an expression : $$ (\alpha A + \beta B)^\top (\alpha A + \beta B) $$ where $\alpha,\beta$ are two real numbers and $A,B$ ...
5
votes
1answer
116 views

How to read off coefficients of tensor-like expression in a speedy way?

I am considering identities involving t[a, b, c, d, ...], where number of indices is fixed. t has the cyclic property so that ...
5
votes
1answer
181 views

Summing tensors in mathematica

How do I perform the following summation in mathematica? \begin{equation} \Sigma_{m=1}^5 e_{ijklm}A^{mn} \end{equation} I have the $e_{ijklm}$ tensor of rank 5 in 5 dimension as a array and $A^{mn}$ ...
5
votes
1answer
226 views

Problems with CircleTimes and infix notation

I am trying to create a function called TensorBasis that takes as input a list (thought of as a list of names of basis vectors of a vector space) and an integer ...
5
votes
0answers
72 views

Can I use symbolic tensors for simple linear algebra and calculus?

As an example, I'd like to calculate this symbolic derivative: $\frac{\partial}{\partial b}(x-A.b)^{\mathsf{T}}.(x-A.b)$ where $x$ and $b$ are vectors and $A$ is a matrix. What I tried is this: ...
4
votes
2answers
248 views

Simplifying the trace of a matrix expression

I have a long expression involving matrices that I derived using the NCAlgebra package. Can I simplify the trace of the expression? For example, say b is a scalar ...
4
votes
1answer
137 views

On elegant use of Inner and Outer on tensors

I have a collection of 5-element vectors each of which corresponds to an {x,y} location from a (20 x 21) grid. Because of that I represent that data as a (5 x 20 x 21)-sized tensor called ...
4
votes
1answer
96 views

Mathematica package for supergravity and superstring theory

I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...
3
votes
3answers
193 views

Matrix multiplication that includes a tensor

How would I best express the following in Mathematica: $\begin{pmatrix}2 & 4\end{pmatrix} \begin{pmatrix}r_1 & r_2\\r_3 & r_4\end{pmatrix} \begin{pmatrix}6 \\ 8\end{pmatrix}$, where $r_i$ ...
3
votes
2answers
345 views

How to solve a “tensor equation”?

I am trying to solve equations which looks like this: $$ T_{ab} - T_{bc} = a_1 T_{ab} + a_2 T_{ac} + a_3 T_{bc}, $$ where $T_{xy}$ are tensors. I want to get the $a_i$'s (in this simple example ...
3
votes
2answers
177 views

Mixed product identity between tensors in Mathematica 9

How can we simplify tensor expressions in Mathematica 9 using the mixed-product identity $(A\otimes B)(C \otimes D) \equiv AC \otimes BD$ ? Is it possible to implement this kind of evaluations using ...
3
votes
1answer
288 views

How can I get Mathematica to recognize equality of symbolic matrix expressions?

I have two matrix expressions: X.Transpose[T].Transpose[X] and X.T.Transpose[X] I want Mathematica to recognize that ...
3
votes
3answers
77 views

How to select elements in a tensor conditional upon their indices

I have a tensor, u, of rank one, meaning that I have a matrix whose elements are themselves matrices. I would like to select, and make a list of, only those sub-matrices whose indices comply with the ...
3
votes
0answers
137 views

Symbolic tensor simplifications and the identity matrix

How can I get Mathematica to simplify the following expressions, with $Assumptions including Element[n,Integers], n > 0, and ...
2
votes
3answers
190 views

Sum with Levi-Civita

I'm trying to write the expression $$\sum_{\alpha,\beta = 1}^{4}\epsilon_{\mu \nu\alpha\beta}a^{\nu} b^{\alpha} c^{\beta}$$ in Mathematica, where $\epsilon$ is the Levi-Civita symbol and $a$, $b$, $c$ ...
2
votes
1answer
121 views

Covariant derivative for symbolic tensors

I want to define a "prefix" (D_i) covariant derivative operator CD[] for symbolic tensors in form of a function, i.e. for ...
2
votes
1answer
458 views

Contracting with Levi-Civita (totally antisymmetric) tensor

I have an array $v_{ijk}$ which is effectively a rank-$3$ tensor with dimensions $3\times3\times3$, and I need to contract it with $e^{ijk}$, i.e. evaluate $v_{ijk} e^{ijk}$ (see Einstein ...
2
votes
0answers
85 views

How to solve system of differential equations of arbitrary order (symbolic tensors)?

I am interested in solving systems of ODEs symbolicly, keeping things with arbitrary dimensions for clarity. For example, assume that $x, f(x) \in R^N$ and $A \in R^{N \times N}$, how do I solve ...
2
votes
0answers
107 views

Prevent Suppression of Superscript 1 in Print

I'm trying to print Christoffel symbols of the second kind for a surface in $\mathbb R^3$. I currently am using something along the lines of ...
2
votes
0answers
115 views

Smooth Max and Abs

I'm trying to implement smooth approximations of Max and Abs functions. Moreover I want the functions to map element-wise on tensors. Here's my code. ...
2
votes
0answers
380 views

how to associate a metric for tensor contraction operations?

I see that Version 9 now has some built in tensor support (which was missing back when previously asked How to represent and manipulate abstract indexed vector (or tensor) expressions?). The docs ...
1
vote
2answers
237 views

Using the epsilon tensor in Mathematica

I'm having a great deal of trouble getting started on a weekly homework assignment in Mathematica. The assignment requires we use the epsilon tensor which is apparently built into Mathematica as ...
1
vote
1answer
84 views

Mathematica does not respect tensor order?

I'm trying to verify an identity involving symmetric traceless tensors over the reals. What I tried was: ...
1
vote
2answers
275 views

Doing vector manipulations on Mathematica

This is hopefully a simpler version of this previous unanswered question of mine. Let me just focus on the two expressions $F_2^{(s)}$ and $F_3^{(s)}$ given in A.3 and A.4 of page 19 of this paper. ...
1
vote
1answer
99 views

Matrix/tensor addition behaving funny under replacement?

I have two matrices I want to add, and one of the matrices is a tensor product of two vectors. I've used a SetDelayed to define the summed matrix, because I want to evaluate it for different values of ...
1
vote
2answers
60 views

Defining tensor components generally

I would like to define a tensor according to its components, something like the following: [F(x)]_i,j = Integrate[f(x,y)A_i(y)B_j(y),y] where i and j are ...
1
vote
1answer
73 views

Why TensorReduce not working here?

I would like to use TensorReduce to work on the tensor contraction $y_{ib}y_{ia}$ (where index $i$ is summed). However, ...
1
vote
1answer
49 views

Evaluating the tensor product many times for a given list

I need to evaluate the tensor product 10 times in a given list( {g,e,r}) Here I have evaluated for 3 times... ...
1
vote
1answer
137 views

Linearized Einstein Equations with Mathematica

I need to compute the linearised Einstein Equations around a fixed metric $g_{\mu \nu}$ which is not the flat metric. Someone knows some Mathematica package or some review that can help me?
1
vote
0answers
39 views

xCoba: evaluate tensor quantities, given an explicit metric

I am new to xAct packages and I'm having some problems to compute tensor quantities with xCoba. I defined a manifold and a metric. I computed All quantities of ...