Tensors are fundamental tools for linear computations, generalizing vectors and matrices to higher ranks. The Wolfram Language includes powerful methods to algebraically manipulate tensors with any rank and symmetry. It handles both tensors given as arrays of components and symbolic tensors given as members of specific tensor domains.

Tensor Representation and Properties

Arrays — domain of symbolic arrays with given properties

Matrices  ▪  Vectors

TensorRank  ▪  TensorDimensions  ▪  TensorSymmetry

Tensor Algebra

TensorContract — contractions of slots of tensors

TensorTranspose — transposition of tensor slots

TensorProduct — general product of tensors

TensorWedge  ▪  HodgeDual  ▪  Symmetrize

Tensor Canonicalization

TensorReduce — convert any polynomial tensor expression into a canonical form

TensorExpand — expand out products, sums, and other tensor operations

Arrays with Symmetry

SymmetrizedArray — array specified by its independent components under symmetry

SymmetrizedArrayRules — rules for independent components of an array with symmetry

SymmetrizedReplacePart — replace independent and corresponding dependent components of an array based on symmetry

Tensor Symmetry Specifications

Symmetric — effect of transposing tensor slots

Antisymmetric  ▪  ZeroSymmetric  ▪  Hermitian  ▪  Antihermitian

SymmetrizedIndependentComponents  ▪  SymmetrizedDependentComponents