Tensors
Tensors are mathematical objects that give generalizations of vectors and matrices. In
Mathematica, a tensor is represented as a set of lists, nested to a certain number of levels. The nesting level is the
rank of the tensor.
rank 0  scalar 
rank 1  vector 
rank 2  matrix 
rank k  rank k tensor 
Interpretations of nested lists.
A tensor of rank
k is essentially a
kdimensional table of values. To be a true rank
k tensor, it must be possible to arrange the elements in the table in a
kdimensional cuboidal array. There can be no holes or protrusions in the cuboid.
The
indices that specify a particular element in the tensor correspond to the coordinates in the cuboid. The
dimensions of the tensor correspond to the side lengths of the cuboid.
One simple way that a rank
k tensor can arise is in giving a table of values for a function of
k variables. In physics, the tensors that occur typically have indices which run over the possible directions in space or spacetime. Notice, however, that there is no builtin notion of covariant and contravariant tensor indices in
Mathematica: you have to set these up explicitly using metric tensors.
Table[f,{i_{1},n_{1}},{i_{2},n_{2}},...,{i_{k},n_{k}}] 
 create an n_{1}×n_{2}×...×n_{k} tensor whose elements are the values of f 
Array[a,{n_{1},n_{2},...,n_{k}}]  create an n_{1}×n_{2}×...×n_{k} tensor with elements given by applying a to each set of indices 
ArrayQ[t,n]  test whether t is a tensor of rank n 
Dimensions[t]  give a list of the dimensions of a tensor 
ArrayDepth[t]  find the rank of a tensor 
MatrixForm[t]  print with the elements of t arranged in a twodimensional array 
Functions for creating and testing the structure of tensors.
This is another way to produce the same tensor.
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MatrixForm displays the elements of the tensor in a twodimensional array. You can think of the array as being a 2×3 matrix of column vectors.
Out[3]//MatrixForm= 
 

Here is the 111 element of the tensor.
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The rank of a tensor is equal to the number of indices needed to specify each element. You can pick out subtensors by using a smaller number of indices.
Transpose[t]  transpose the first two indices in a tensor 
Transpose[t,{p_{1},p_{2},...}]  transpose the indices in a tensor so that the k^{th} becomes the p_{k}^{th} 
Tr[t,f]  form the generalized trace of the tensor t 
Outer[f,t_{1},t_{2}]  form the generalized outer product of the tensors t_{1} and t_{2} with "multiplication operator" f 
t_{1}.t_{2}  form the dot product of t_{1} and t_{2} (last index of t_{1} contracted with first index of t_{2}) 
Inner[f,t_{1},t_{2},g]  form the generalized inner product, with "multiplication operator" f and "addition operator" g 
Tensor manipulation operations.
You can think of a rank
k tensor as having
k "slots" into which you insert indices. Applying
Transpose is effectively a way of reordering these slots. If you think of the elements of a tensor as forming a
kdimensional cuboid, you can view
Transpose as effectively rotating (and possibly reflecting) the cuboid.
In the most general case,
Transpose allows you to specify an arbitrary reordering to apply to the indices of a tensor. The function
Transpose[T, {p_{1}, p_{2}, ..., p_{k}}], gives you a new tensor
T^{} such that the value of
T^{}_{i1 i2 ... ik} is given by
T_{ip1 ip2 ... ipk}.
If you originally had an
n_{p1}×
n_{p2}×
...×
n_{pk} tensor, then by applying
Transpose, you will get an
n_{1}×
n_{2}×
...×
n_{k} tensor.
Here is a matrix that you can also think of as a 2×3 tensor.
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The element m[[2, 3]] in the original tensor becomes the element m[[3, 2]] in the transposed tensor.
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This produces a 2×3×1×2 tensor.
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This transposes the first two levels of t.
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The result is a 3×2×1×2 tensor.
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If you have a tensor that contains lists of the same length at different levels, then you can use
Transpose to effectively collapse different levels.
This collapses all three levels, giving a list of the elements on the "main diagonal".
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This collapses only the first two levels.
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You can also use
Tr to extract diagonal elements of a tensor.
This forms the ordinary trace of a rank 3 tensor.
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Here is a generalized trace, with elements combined into a list.
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This combines diagonal elements only down to level 2.
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Outer products, and their generalizations, are a way of building higherrank tensors from lowerrank ones. Outer products are also sometimes known as direct, tensor or Kronecker products.
From a structural point of view, the tensor you get from
Outer[f, t, u] has a copy of the structure of
u inserted at the "position" of each element in
t. The elements in the resulting structure are obtained by combining elements of
t and
u using the function
f.
This gives the "outer f" of two vectors. The result is a matrix.
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If you take the "outer f" of a length 3 vector with a length 2 vector, you get a 3x2 matrix.
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The result of taking the "outer f" of a 2x2 matrix and a length 3 vector is a 2x2x3 tensor.
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Here are the dimensions of the tensor.
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If you take the generalized outer product of an
m_{1}×
m_{2}×
...×
m_{r} tensor and an
n_{1}×
n_{2}×
...×
n_{s} tensor, you get an
m_{1}×
...×
m_{r}×
n_{1}×
...×
n_{s} tensor. If the original tensors have ranks
r and
s, your result will be a rank
r+s tensor.
In terms of indices, the result of applying
Outer to two tensors
T_{i1 i2 ... ir} and
U_{j1 j2 ... js} is the tensor
V_{i1 i2 ... irj1 j2 ... js} with elements
f[T_{i1 i2 ... ir}, U_{j1 j2 ... js}].
In doing standard tensor calculations, the most common function
f to use in
Outer is
Times, corresponding to the standard outer product.
Particularly in doing combinatorial calculations, however, it is often convenient to take
f to be
List. Using
Outer, you can then get combinations of all possible elements in one tensor, with all possible elements in the other.
In constructing
Outer[f, t, u] you effectively insert a copy of
u at every point in
t. To form
Inner[f, t, u], you effectively combine and collapse the last dimension of
t and the first dimension of
u. The idea is to take an
m_{1}×
m_{2}×
...×
m_{r} tensor and an
n_{1}×
n_{2}×
...×
n_{s} tensor, with
m_{r}=n_{1}, and get an
m_{1}×
m_{2}×
...×
m_{r1}×
n_{2}×
...×
n_{s} tensor as the result.
The simplest examples are with vectors. If you apply
Inner to two vectors of equal length, you get a scalar.
Inner[f, v_{1}, v_{2}, g] gives a generalization of the usual scalar product, with
f playing the role of multiplication, and
g playing the role of addition.
This gives a generalization of the standard scalar product of two vectors.
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This gives a generalization of a matrix product.
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Here are the dimensions of the result.
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You can think of
Inner as performing a "contraction" of the last index of one tensor with the first index of another. If you want to perform contractions across other pairs of indices, you can do so by first transposing the appropriate indices into the first or last position, then applying
Inner, and then transposing the result back.
In many applications of tensors, you need to insert signs to implement antisymmetry. The function
Signature[{i_{1}, i_{2}, ...}], which gives the signature of a permutation, is often useful for this purpose.
Outer[f,t_{1},t_{2},...]  form a generalized outer product by combining the lowestlevel elements of t_{1}, t_{2}, ... 
Outer[f,t_{1},t_{2},...,n]  treat only sublists at level n as separate elements 
Outer[f,t_{1},t_{2},...,n_{1},n_{2},...]  treat only sublists at level n_{i} in t_{i} as separate elements 
Inner[f,t_{1},t_{2},g]  form a generalized inner product using the lowestlevel elements of t_{1} 
Inner[f,t_{1},t_{2},g,n]  contracts index n of the first tensor with the first index of the second tensor 
Treating only certain sublists in tensors as separate elements.
Here every single symbol is treated as a separate element.
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But here only sublists at level 1 are treated as separate elements.
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Flattening block tensors.
Here is a block matrix (a matrix of matrices that can viewed as blocks that fit edge to edge within a larger matrix).
Out[30]//TableForm= 
 
