1.2.67 Tensor Component Symmetry Definition
Tensor component symmetry defines how tensor elements transform under permutations, revealing invariance properties critical in physics and mathematics.
Tensor Component Symmetry Definition is the specification of the invariance a tensor's components exhibit when two or more of its indices are exchanged, classifying tensors according to whether such an exchange leaves the components unchanged, reverses their sign, or produces no consistent relationship at all. Symmetry is a property defined among indices of the same type, either exclusively among the upper indices or exclusively among the lower indices, since exchanging an upper index with a lower index is not a meaningful operation within the standard tensor formalism.
Symmetric Components
Definition
A tensor is symmetric in a specified pair of indices of the same type when exchanging the values assigned to that pair leaves the component value unchanged.
for every choice of values assigned to i and j and to any other indices held fixed. If a tensor is symmetric in every pair among a full set of indices, it is called totally symmetric in those indices.
Consequence for the Component Grid
For a rank-two tensor symmetric in its two indices, the component grid is a mirror image of itself across the main diagonal, so that the entry in row i, column j always equals the entry in row j, column i.
Antisymmetric Components
Definition
A tensor is antisymmetric, also called skew-symmetric, in a specified pair of indices of the same type when exchanging the values assigned to that pair reverses the sign of the component value.
If a tensor is antisymmetric in every pair among a full set of indices, it is called totally antisymmetric in those indices, and such tensors are the algebraic foundation for differential forms.
Forced Vanishing of Diagonal Entries
Setting i = j in the antisymmetry relation gives T_{...i...i...} = -T_{...i...i...}, which has only the solution T_{...i...i...} = 0, so every component with a repeated index among an antisymmetric pair must equal zero.
Symmetry Under a Change of Basis
Preservation of Symmetry Type
If a tensor's components are symmetric or antisymmetric in a particular basis, the same symmetry or antisymmetry holds in every other basis, since the transformation law applies the identical change-of-basis factors to both the original component and its exchanged counterpart, preserving whatever equality or sign relation existed between them. Symmetry is therefore a basis-independent property of the abstract tensor, not an accident of a particular coordinate choice.
Decomposition into Symmetric and Antisymmetric Parts
Splitting a General Rank-Two Tensor
Any rank-two tensor with both indices of the same type can be decomposed uniquely into the sum of a symmetric part and an antisymmetric part.
where the symmetric part and antisymmetric part are defined explicitly by
The parentheses notation T_{(ij)} denotes symmetrization and the square bracket notation T_{[ij]} denotes antisymmetrization, standard shorthand conventions used throughout tensor algebra.
Tensors Without Definite Symmetry
Mixed Type Indices Cannot Be Compared
Symmetry statements only apply to pairs of indices of the same variance; a component such as T^i_j in a type (1, 1) tensor cannot be meaningfully called symmetric or antisymmetric with respect to exchanging i and j, since one index is contravariant and the other covariant, and they do not transform the same way under a change of basis.
General Tensors with No Symmetry
Many tensors possess no symmetry at all among any pair of their indices of matching type, and their components must be specified individually without the reduction in independent entries that symmetric or antisymmetric structure provides.
Diagrammatic Summary
The diagram contrasts symmetric index exchange, where swapping two indices leaves the component value unchanged, with antisymmetric index exchange, where swapping the same two indices reverses the sign of the component value.