1.2.66 Tensor Redundant Component Definition

Tensor Redundant Component Definition is the specification of the entries within a tensor's component array whose values are not free to be chosen independently but are instead algebraically determined by other components through symmetry, antisymmetry, or additional identities that the tensor satisfies. A redundant component is the complement of an independent component: it is included in the raw component array of size n^(p+q), yet it contributes no new information beyond what is already fixed by the tensor's independent components and the relations that link them.

Redundancy as the Complement of Independence

Counting Redundant Components

If a tensor has n^(p+q) raw components in total and a smaller number of independent components determined by its symmetry structure, the redundant components are exactly the difference between these two counts.

For a symmetric rank-two tensor over an n-dimensional space, the independent count is n(n+1)/2, so the redundant count is the remainder.

which is exactly the number of off-diagonal entries below the main diagonal, each one equal in value to its mirrored counterpart above the diagonal.

Sources of Redundancy

Redundancy from Symmetry Relations

When a tensor is symmetric in a pair of indices, T_{ij} = T_{ji}, one member of every off-diagonal pair is redundant, since its value is already fixed once the other member of the pair is known. The choice of which member of the pair to regard as independent and which as redundant is a matter of convention, not a structural distinction.

Redundancy from Antisymmetry Relations

When a tensor is antisymmetric in a pair of indices, T_{ij} = -T_{ji}, every diagonal entry is redundant in the strongest possible sense, forced to equal exactly zero rather than merely being tied to another entry's value, and one member of every off-diagonal pair is redundant as well, determined as the negative of the other.

Redundancy from Additional Algebraic Identities

Some tensors satisfy identities beyond simple index-pair symmetry, such as cyclic identities relating three or more indices at once. These identities introduce further redundancy among the entries that a naive symmetry count alone would classify as independent, requiring a more careful accounting of the full set of constraints to determine the true number of independent components.

Why Redundant Components Still Appear in the Array

Convenience of a Uniform Index Structure

Although redundant components carry no independent information, tensor components are conventionally still written with the full set of p + q indices, each ranging over the complete set of values, rather than restricted to only the independent entries. This uniform indexing keeps the transformation law, summation conventions, and algebraic manipulations consistent and simple, even at the cost of storing values that could in principle be derived from others.

Practical Consequences for Computation

In numerical or symbolic computation, recognizing redundant components allows for more efficient storage, since only the independent components need to be recorded explicitly, with redundant entries reconstructed on demand using the known symmetry relation. Ignoring redundancy, by contrast, means unnecessarily storing and manipulating values that add no new content to the description of the tensor.

Example: The Metric Tensor

Redundancy in a Symmetric Metric

A metric tensor is a type (0, 2) tensor that is symmetric, g_{ij} = g_{ji}, so in a four-dimensional space it has 16 raw components but only 10 independent components, the 6 off-diagonal entries below the diagonal being redundant, each equal to its counterpart above the diagonal.

Diagrammatic Summary

The diagram marks the lower-triangular region of a symmetric grid as redundant, each entry there being a mirror copy of the corresponding entry in the upper triangle, while the diagonal and upper triangle together contain the tensor's independent components.