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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1answer
72 views

Tensors constructed using KroneckerDelta's - and/or displaying KroneckerDelta as a matrix

I am doing some work in elasticity and as a result am working with tensors. In particular, I would like to calculate the contraction of a fourth order tensor (the stiffness tensor) with a second order ...
4
votes
1answer
85 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 ...
3
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1answer
231 views

SymmetrizedArray of stiffness/compliance tensor

The stiffness and compliance tensors of a material are 3x3x3x3 tensors relating stress and strain. ...
1
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1answer
43 views

Efficient implementation of tensorial Rayleigh product

I am interested in the tensor product $\hat{B} = A \star B$ (which at least I know as Rayleigh product), defined with components \begin{equation} \hat{B}_{i_1 i_2 ... i_ n} = \sum_{j_1 = 1}^d \sum_{...
3
votes
1answer
84 views

Finding basis of isotropic tensors of rank $n$

I am looking for a way to obtain a basis of isotropic tensors of rank $n$. Actually I am mostly interested in rank $8$ isotropic tensors but maybe you know already a simple algorithm in order to ...
1
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2answers
101 views

Output the tensor product of two matrix as a matrix

How do I output the matrix form like the RHS without the tensor product sign remaining $\otimes$? I need it for display purpose where I can see easily what the form of the whole product matrix is.
0
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0answers
63 views

Scalar from tensor contraction

I'm trying to calculate the Kretschmann scalar in mathematica, it is given by: $c = R^{abcd} R_{abcd}$ Where $R^{abcd}$ is the Riemann tensor. I'm following this MSE post so I modified it to ...
0
votes
1answer
58 views

How to multiply two tensor with arbitrary ranks, on one index only (like GR)?

I am writing a function which take two tensors with the same dimensions but with arbitrary ranks and multiplies them over one index, just like this example from GR: See, I couldn't use ...
0
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1answer
22 views

Dynamically constructing symbolic tensors

Is there any way to construct tensor of any given dimension d and rank r with symbolic entries? i.e., instead of manually constructing one as Table[T[i,j,k],{i,1,10},{i,1,10},{i,1,10}] for d=3 and r=...
3
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1answer
53 views

Constructing tensors with arbitrary rank and dimension [closed]

How can I create a tensor with arbitrary rank n and m components in each dimension? I am looking for a command in which I insert n and m and it spits out the corresponding tensor with random entries (...
2
votes
1answer
77 views

Working with tensor algebra

My question is really easy for experienced users. In my tensor equations I have an unknown tensor Q (symmetric and traceless): $Q=\begin{pmatrix} n1(x,y) & n2(x,y) \\ n2(x,y) & -n1(x,y) \end{...
1
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0answers
107 views

Outer product using the quantum mathematica package

I am using the quantum Mathematica package and am trying to do computations that require me to use the outer (tensor) products of Pauli matrices. I would define this as, for example, ...
3
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2answers
91 views

Perform matrix/tensor contractions more efficiently

Problem: explicit index contractions for matrices using the Sum command take too long, and I want to improve the performance for complicated computations. Let me ...
8
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0answers
166 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 ...
0
votes
1answer
106 views

How to create simple (tensor) product spaces?

Is it possible to work with simple tensor product spaces, like multiplying product states from quantum mechanics? I basically have a simple two dimensional vector space, whose elements are ...
15
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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 ...
3
votes
2answers
80 views

Find most general 3-tensor under given constraint?

Consider A to be a general $2\times 2\times 2$ tensor: A = Table[ a[i,j,k], {i, 1, 2}, {j, 1, 2}, {k, 1, 2}]; pictorially one ...
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. ...
1
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2answers
124 views

Update: Calculate Tensor products and traces with Mathematica

My goal is to algebraically simplify expressions involving tensor products and taking the trace. That is, I would like to compute $\operatorname{Tr}\left( \prod_i(id+a_i \otimes b_i) \right)$, using ...
2
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0answers
60 views
4
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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: ...
0
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1answer
82 views
3
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1answer
117 views

Expanding the Riemann tensor (perturbation)

I am trying to expand the following term (via the perturbation expansion $g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}$): $$ \frac12 \epsilon^{abcd}R^e_{fcd}R^f_{eab} $$ Where $R$ is the Riemann ...
2
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0answers
100 views

Fast construction and reshaping of large tensors

I'm trying to construct a large matrix which is derived from some higher rank tensor (the rank of interest to me changes case by case, so it needs to be a general method). Currently, the process of ...
2
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1answer
128 views

Explicit Contraction of Tensor Indices [duplicate]

Is there any easy way to explicitly contract indices of several given tensors. For example, ...
6
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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: ...
3
votes
1answer
63 views

How to fill a tensor with matrices one by one?

I have this tensor: nn = 4; tableToFill = Table[x, {p, 1, nn}, {q, 1, nn}, {i, 1, nn}, {j, 1, nn}]; that I want to fill with a matrix one by one with a matrix ...
0
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0answers
325 views

Symbolic tensor product calculation

I am trying to calculate the a symbolic tensor product $\langle \nabla f\otimes f,\nabla g\otimes g\rangle ^2+(\langle \nabla f\otimes g,\nabla f\otimes g\rangle +\langle \nabla g\otimes f,\nabla g\...
6
votes
1answer
74 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}....
5
votes
1answer
510 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?
0
votes
1answer
192 views

Nonrectangular tensor encountered [closed]

I am trying to solve a system of equations which characteristic matrix: ...
1
vote
2answers
93 views

Displaying tensors

I want to display the values of a table with superscirpts and subscripts in a list as bellow. ...
3
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0answers
57 views

Nested TensorContract bug

Bug introduced in version 9.0.0 or 9.0.1 and fixed in 10.1 or 10.2 When nesting TensorContract the result I get is wrong. Input: ...
1
vote
1answer
89 views

Calculation of tensor structure

I would like to calculate following type of contraction: $$ P_{ij}^{p-k} P_{lm}^{k} ((p_l - k_l) \delta_{ij} + k_j \delta_{il}) (k_n \delta_{im} - p_m\delta_{in}) = ? $$ where p,k are d-...
1
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0answers
84 views

Ricci tensor with Mathematica [closed]

I'm trying for first time to write a Ricci tensor with Mathematica. The code I've written is this: ...
2
votes
2answers
87 views

Syntax for higher rank tensor multiplication

I am trying to do some matrix multiplication in Mathematica but I just cannot figure out the correct syntax for my problem. I want to write the following: $$ A\pmatrix{a & b\\c & d}+B\pmatrix{...
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 ...
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 ...
1
vote
2answers
409 views

Compute a double dot product between two tensors of rank 3 and 2

I would need help to calculate a double dot product between a rank 3 tensor A and a rank 2 tensor B (A:B) using mathematica. Does someone know how to do that? Thank you for your help!
2
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0answers
59 views

Simplifying symbolic expressions using TensorExpand

Following my previous question I have this issue using TensorExpand: KroneckerProduct[x, y].(KroneckerProduct[2 z, w]) // TensorExpand results in ...
4
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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. ...
2
votes
1answer
98 views

Choosing Minkowski metric in Feyncalc

This is related to the Mathematica package FeynCalc. Is there a way to simplify the metric? I mean in the sense that when I evaluate something and I get a long expression involving lots of $$g^{12}$$...
1
vote
1answer
111 views

Curl of a second-order tensor

In Mathematica 9.0, the documentation for the Curl function states that in n-dimensions "the resulting curl is an array with depth n-k-1 of dimensions". Accordingly, if a 2-dimensional array is feeded ...
2
votes
0answers
57 views

Arrays with two different types of indices

Is there a way of having arrays which have, so to speak, two different depths $m,n$? From the point of view of memory usage, it would be the same as an array of depth $m+n$, but I would like to ...
3
votes
1answer
44 views

Simplifying nested KroneckerProducts

Any suggestion how to bring a nested KroneckerProduct form into just one? I mean some how converting ...
1
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0answers
60 views

Continuous integration of a tensor function using discrete density values

Hei, I am wondering is it possible to implement continuous integration using Integrate() of a fourth order tensor with discrete density values. ...
13
votes
1answer
241 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 ...
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 ...
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 ...
1
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1answer
78 views

Issue with Differentiation over a tensor

I am new user in Mathematica. I am working with liquid crystal theory, and one of the notation there is the tensor: $ \Pi_{ij} = \frac {\partial {F_{el}}} { \partial ( \partial {n_i} / \partial { x_j}...