TensorProduct

TensorProduct[tensor₁,tensor₂,…]

represents the tensor product of the tensor_i.

Details

TensorProduct[a,b] can be input as ab. The character  is entered as t* or \[TensorProduct].
The tensor product a₁…a_n of rectangular arrays a_i is equivalent to Outer[Times, a₁,…,a_n].
The tensor product t₁…t_n of arrays and/or symbolic tensors is interpreted as another tensor of rank TensorRank[t₁]+…+TensorRank[t_n].
TensorProduct[] returns 1. TensorProduct[x] returns x.
TensorProduct is an associative, non-commutative product of tensors.

Examples

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Basic Examples (2)Summary of the most common use cases

Tensor product of arrays:

Out[1]=1

Tensor product of symbolic expressions:

Out[1]=1

Expand linearly:

Out[2]=2

Compute properties of tensorial expressions:

Out[3]=3

Scope (4)Survey of the scope of standard use cases

Tensor product of arrays of any depth and dimensions:

Out[1]=1

Out[2]=2

Out[3]=3

Out[4]=4

Product of symmetrized arrays, with the result also in symmetrized form:

Out[1]=1

Out[2]=2

Out[3]=3

The fact that both arrays are the same adds more symmetry:

Out[4]=4

There are only six nonzero independent components:

Out[5]=5

Tensor product of symbolic expressions:

Out[1]=1

Out[2]=2

Out[3]=3

Tensor product of objects of different types. Contiguous arrays are multiplied:

Out[1]=1

Properties & Relations (11)Properties of the function, and connections to other functions

The tensor product is not commutative:

Out[1]=1

Out[2]=2

The difference is always some transposition:

Out[3]=3

The tensor product of arrays is equivalent to the use of Outer:

Out[1]=1

Out[2]=2

Out[3]=3

The KroneckerProduct of vectors is equivalent to their TensorProduct:

Out[1]=1

The KroneckerProduct of matrices is equivalent to the flattening of their TensorProduct to another matrix:

Out[2]=2

The KroneckerProduct of any two arrays is also equivalent to a flattening of their TensorProduct:

Out[3]=3

The rank of a tensor product is the sum of ranks of the factors:

Out[1]=1

Out[2]=2

The tensor product of a tensor with itself gives a result with added symmetry:

Out[2]=2

TensorProduct[x] returns x irrespectively of what x is:

Out[1]=1

TensorProduct[] is 1:

Out[1]=1

Obvious scalars are extracted from a tensor product:

Out[1]=1

Symbolic scalars need to be specified with assumptions:

Out[2]=2

TensorProduct has Flat attribute:

Out[2]=2

TensorProduct, in combination with TensorContract, can be used to implement Dot:

Out[2]=2

Antisymmetrization of TensorProduct is proportional to TensorWedge:

Out[1]=1

Out[2]=2

Out[3]=3

Out[4]=4

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