7.16.2 Tensor Component Symmetric Equality Rule

Tensor Component Symmetric Equality Rule is the rule stating that for a symmetric tensor, any two components obtained from one another by exchanging a designated pair of indices are equal in value, for every choice of the remaining indices and in every admissible coordinate system. This rule is what defines symmetry as a property of the components of a tensor, and it holds as an identity rather than as a coincidence that depends on a particular basis.

Statement of the Rule

Basic Form for a Rank-Two Tensor

For a covariant rank-two tensor with components T subscript i j, the Tensor Component Symmetric Equality Rule states that the tensor is symmetric in its two indices precisely when:

for every value of the indices i and j. The same statement applies equally to a contravariant tensor with components T superscript i j, where the rule reads:

Extension to Higher Rank Tensors

For a tensor of rank greater than two, the rule is applied to a specified pair of indices while all other indices are held fixed. For a rank-three tensor symmetric in its first two indices, the rule takes the form:

Symmetry in additional pairs of indices, or symmetry in every possible pair simultaneously, is described by applying this equality rule to each pair in turn.

Basis Independence of the Rule

Why Symmetry Survives a Change of Coordinates

A key feature of the Tensor Component Symmetric Equality Rule is that it is preserved under an admissible change of coordinates. If the equality holds for the components of a tensor in one coordinate system, the transformation law for tensor components guarantees that it also holds in every other coordinate system. This follows because the transformation of each index uses the same set of partial derivative factors regardless of the order in which the indices happen to be written, so exchanging the indices before or after the transformation produces the same result.

Symmetry as a Property of the Object, Not the Representation

Because the equality survives every admissible change of coordinates, symmetry described by this rule is a property of the tensor as an invariant object rather than an accident of a particular set of components. A tensor cannot be symmetric in one coordinate system and asymmetric in another; the property is intrinsic.

Illustration

The diagram represents two entries of a symmetric tensor's component array positioned symmetrically across the diagonal. The equality rule asserts that these two entries always carry the same value.

Consequences of the Rule

Reduction in the Number of Independent Components

A direct consequence of the Tensor Component Symmetric Equality Rule is that a symmetric tensor has fewer independent components than a general tensor of the same rank, since every pair of components related by the rule counts as a single independent value rather than two. For a rank-two symmetric tensor in an n-dimensional space, the number of independent components is reduced from n squared to n times the quantity n plus one, divided by two.

Compatibility With Contraction and Linear Combination

If two tensors both satisfy the Tensor Component Symmetric Equality Rule with respect to the same pair of indices, any linear combination of those tensors also satisfies the rule with respect to that pair. Symmetry is therefore preserved under addition and scalar multiplication, which allows symmetric tensors of a given rank to be treated as a well-defined subset within the space of all tensors of that rank.

Interaction With Antisymmetric Components

Any tensor can be decomposed into a part that obeys the Tensor Component Symmetric Equality Rule and a part that obeys the corresponding antisymmetric equality rule, in which exchanging the indices reverses the sign of the component rather than preserving it. This decomposition is unique, and it separates the symmetric behavior described here from the strictly opposite behavior found in antisymmetric tensors.

Relationship to Other Tensor Concepts

The Tensor Component Symmetric Equality Rule is the defining member of the broader Tensor Component Symmetry Pattern, which also includes antisymmetric equality rules and mixed symmetry patterns applying to more than two indices at once. It is closely tied to Tensor Component Object Preservation, since the invariance of the equality rule under coordinate change is itself a specific instance of the general principle that a tensor's intrinsic properties are preserved regardless of how its components are expressed.