Function Repository Resource:

IdentityTensorReduce

Simplify expressions containing a symbolic identity tensor

Contributed by: Carl Woll

ResourceFunction["IdentityTensorReduce"][expr]

converts TensorContract expressions containing symbolic identity tensors into equivalent forms with the identity tensors removed.

Details and Options

ResourceFunction["IdentityTensorReduce"] makes no assumptions about the rank of the symbolic tensors involved. Use the Assumptions option or $Assumptions to define rank information for the involved tensors.

The input and output of ResourceFunction["IdentityTensorReduce"] will evaluate to the same array when given explicit arrays as input.

Examples

Basic Examples (3)

A simple tensor contraction of a tensor with an identity matrix:

In[1]:=

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TensorReduce does not simplify the tensor:

In[2]:=

$TensorReduce[e, Assumptions -> X \[Element] Matrices[{n, n}]]$

Out[2]=

IdentityTensorReduce simplifies the tensor:

In[3]:=

$ResourceFunction["IdentityTensorReduce"][e, Assumptions -> X \[Element] Matrices[{n, n}]]$

Out[3]=

Use Inactive when the dimensions of the identity tensor are explicit:

In[4]:=

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IdentityTensorReduce simplifies the tensor:

In[5]:=

$ResourceFunction["IdentityTensorReduce"][e, Assumptions -> X \[Element] Matrices[{n, n}]]$

Out[5]=

A tensor expression involving SymbolicIdentityArray:

In[6]:=

$A = MatrixSymbol["A", {d, d}]; e = TensorContract[ A\[TensorProduct]A\[TensorProduct]A\[TensorProduct]SymbolicIdentityArray[{d, d}], {{1, 5}, {2, 4}, {3, 9}, {6, 10}}]$