7.16.5 Tensor Component Symmetric Tensor Role

Tensor Component Symmetric Tensor Role is the function that a tensor exhibiting symmetric behavior in all of its indices plays within tensor algebra, serving as the standard representative object used whenever a quantity must treat its indices interchangeably and must therefore be describable without reference to any particular ordering among them.

What the Role Requires

Full Symmetry Across All Indices

A tensor fulfilling the Symmetric Tensor Role satisfies the Tensor Component Symmetric Equality Rule for every pair of indices of matching variance type, not merely for one designated pair. For a rank-two tensor this means:

for all i and j, while for a rank-three tensor fulfilling the role, the equality must hold under every possible permutation of the three indices, not only under a single exchange.

Total Symmetrization of an Arbitrary Tensor

Any tensor, symmetric or not, can be assigned a corresponding object that fulfills the Symmetric Tensor Role by averaging its components over every permutation of the relevant indices. For a rank-two tensor with components T subscript i j, the symmetrized object is:

This averaged object always satisfies the symmetric equality condition regardless of whether the original tensor did, and it is the standard construction used whenever a symmetric counterpart to an arbitrary tensor is required.

Illustration

Positions the Role Occupies in Tensor Algebra

Representing Quantities With No Preferred Index Order

A tensor fulfilling the Symmetric Tensor Role is the natural object used to represent any quantity whose defining relationship makes no distinction between the roles of its indices. Because such a quantity cannot depend on the order in which its indices are listed, its components must satisfy the symmetric equality condition, and any correctly constructed representation of it will automatically fulfill this role.

Serving as a Building Block in Decomposition

Every tensor of rank two can be written as the sum of a part fulfilling the Symmetric Tensor Role and a part exhibiting the Tensor Component Antisymmetry Pattern. In this decomposition, the tensor fulfilling the Symmetric Tensor Role captures all of the information in the original tensor that is unaffected by exchanging its two indices, while the antisymmetric part captures the remaining information that reverses sign under such an exchange.

Persistence of the Role Under Coordinate Change

The Role Is Not Frame Dependent

A tensor that fulfills the Symmetric Tensor Role in one coordinate system continues to fulfill it in every coordinate system reachable by an admissible transformation, since the symmetric equality condition defining the role is itself preserved under such transformations. A tensor cannot fulfill this role only in special coordinates and fail to fulfill it elsewhere.

Consistency of the Symmetrization Construction

The construction that produces a symmetric counterpart from an arbitrary tensor, by averaging over index permutations, commutes with a change of coordinates. Symmetrizing the components first and then transforming to a new coordinate system yields the same result as transforming first and then symmetrizing, so the symmetric counterpart of a tensor is itself a well-defined, coordinate-independent object.

Relationship to Other Tensor Concepts

Tensor Component Symmetric Tensor Role builds directly on the Tensor Component Symmetric Equality Rule and the broader Tensor Component Symmetry Pattern, extending the notion of a single symmetric index pair to the case where every relevant pair of indices satisfies the same condition simultaneously. It stands in direct contrast to the corresponding role played by fully antisymmetric tensors within the Tensor Component Antisymmetry Pattern.