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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55
votes
2answers
3k views

Internal`Bag inside Compile

Since Internal`Bag, Internal`StuffBag and Internal`BagPart can be compiled down, it is a ...
25
votes
4answers
11k 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. Sometimes,...
16
votes
2answers
3k 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) ...
15
votes
2answers
1k 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 ...
13
votes
1answer
240 views

Nearest Kronecker Product

Some people (see The ubiquitous Kronecker product by Van Loan) have worked on finding two matrices $\mathbf A$,$\mathbf B$ of specified size whose tensor product $\mathbf A\otimes\mathbf B$ is closest ...
11
votes
5answers
2k 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 ...
11
votes
3answers
423 views

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

I would like to make the following matrix: ...
11
votes
1answer
491 views

How can I define or use a new coordinate system in Mathematica?

I want to use the dipole coordinate system as defined in this paper: http://arxiv.org/abs/physics/0606044 I see that Mathematica can do all kinds of vector analysis using different kinds of ...
10
votes
1answer
1k 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 ...
10
votes
3answers
395 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}] ...
10
votes
2answers
563 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 ...
9
votes
1answer
3k 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, ...
9
votes
2answers
304 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 <...
9
votes
1answer
382 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
0answers
153 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: ...
8
votes
5answers
1k 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 + 1,...
8
votes
1answer
336 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 ...
8
votes
1answer
331 views

How to define an orthogonal basis in the right way?

I am trying to work with the vector notation without defining vector components explicitly. $Assumptions = (x | y | z) \[Element] Vectors[3] The vectors ...
8
votes
1answer
291 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 $I_3$? ...
8
votes
0answers
165 views

Efficient way to flatten or transpose-arrayreshape tensors

I have a tensor of dimensions $(2, 100, 100, 2, 100, 100)$ and I want to reshape it to a form of $(2*100*100,2*100*100)$, e.g. Flatten[A,{{1,5,6},{4,2,3}}]. If I ...
7
votes
2answers
368 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 ...
7
votes
2answers
1k views

Perturbation theory in general relativity using xAct

I'm trying to use the xAct Mathematica package for manipulating tensors, and I'd like to plug in a metric into the perturbation equations to first order in general relativity, and have everything ...
7
votes
1answer
262 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 <...
7
votes
1answer
629 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 ...
7
votes
1answer
432 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
3answers
270 views

How to simplify symbolic expressions with KroneckerProduct

Having $X,Y$ being symbols for matrices, I was wondering if there is a way to simplify expressions like KroneckerProduct[X, X] + KroneckerProduct[-X, X] to ...
6
votes
1answer
73 views

Matrix Multiplication after “Flatten”

Say we originally had two matrices, $A$ and $B$, both $n \times n$, whose product $$ C= A.B $$ Now I flatten $A$. I can obtain a flatten $C$, from the following multiplication $$ C_{flat}=A_{flat}....
6
votes
1answer
522 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 $S$ ...
6
votes
1answer
83 views

Issue with TensorWedge and TensorReduce?

Bug introduced in 10.0.0 and fixed in 10.3.1 If I use TensorReduce on the result of TensorWedge: ...
6
votes
1answer
106 views

Error messages from TensorContract and TensorReduce

I am struggling with a few errors when using symbolic tensors. I am using mathematica 9.0.1.0, linux x86. The following code generates what seems to me an incorrect tensor, this is the smallest ...
5
votes
3answers
934 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
1answer
439 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
2answers
444 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 (...
5
votes
1answer
280 views

FindFit returns “tensors have incompatible shapes”

I'm trying to fit experimental data with FindFit and implicitly calculated function. It seems, that function works fine and satisfactorily approximates data when parameters are fed manually. However ...
5
votes
2answers
150 views

The fastest way of raising indices of high rank tensor

I have some high dimensional high rank tensors, let's say $$F_{ijkl}$$ and I need to find $$F^{abcd}=g^{ai}g^{bj}g^{ck}g^{dl}F_{ijkl}.$$ Here $g^{ij}$ is the contravariant metric. Simple summation ...
5
votes
1answer
2k 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 convention)....
5
votes
1answer
508 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 any Mathematica package or a review that can help me?
5
votes
1answer
247 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
259 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 <...
4
votes
1answer
69 views

Creating a tensor from matrices

I have 3 nxn matrices that I'd like to combine into a nxnx3 tensor. Basically, I want a new nxn matrix where each element is a 3 tuple of the elements of the other 3 matrices at that same position. ...
4
votes
2answers
154 views

Using LeviCivitaTensor

I have 2 questions about using LeviCivitaTensor, based on the following session: The signs for the cross product seems to come out "opposite" of what I expected. Why is that? Did I miss anything? ...
4
votes
1answer
113 views

TensorReduce not simplifying transpose for symmetric matrix?

I know Mathematica is not very good with symbolic manipulation of matrix expressions, but I was surprised to find that it can't reduce the following expression: ...
4
votes
1answer
293 views

How to put the tensor product of two operators onto two variables?

I am trying to make an intuitive Mathemtaica program to show the principle of quantum walking. The idea of quantum walking is very simple and direct, but when I try to transfer it into Mathemitca ...
4
votes
1answer
392 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 ...
4
votes
1answer
219 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
42 views

What do I need to know about simplifying expressions involving Symbolic Tensors?

I want to use Mathematica to show that the inner product of a vector with itself is equal to the square of its norm. This is what I tried: ...
4
votes
1answer
84 views

Subscripted (or superscripted) variables in Mathematica

When I first learned of Mathematica and started to use it, I soon discovered that Mathematica supported subscripted and superscripted variables such as $M_{i j}$. Naively, I first thought that I ...
4
votes
0answers
75 views

How to write Coordinate Chart in Xcoba in index form?

I am using xCoba for manipulating tensors. I do the usual, defining my metric, manifold, chart etc. ...
3
votes
1answer
618 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 ...