7.18.3 Tensor Component Dimension Dependence

Tensor Component Dimension Dependence is the fact that the number of components a tensor possesses, whether counted by the raw Tensor Component Total Entry Calculation or by a reduced count arising from symmetry, changes with the dimension of the vector space on which the tensor is defined, and does so in a manner determined precisely by the rank of the tensor and by whichever symmetry pattern its indices may follow.

The Nature of the Dependence

Raw Entry Count as a Function of Dimension

For a tensor of fixed rank r, the Tensor Component Total Entry Calculation expresses the raw entry count as n raised to the power r, where n is the dimension of the underlying space. Holding the rank fixed and varying the dimension shows that the entry count grows according to a power function of n, so that doubling the dimension multiplies the raw entry count by two raised to the power r rather than by a fixed constant amount.

Reduced Entry Count as a Function of Dimension

When a symmetric or antisymmetric relationship is present among a tensor's indices, the reduced component count likewise varies with dimension, but according to a different expression. For a rank-two tensor symmetric in both indices, the independent component count is:

while for a rank-two tensor antisymmetric in both indices, the independent component count is:

Both expressions depend on the dimension n through a quadratic relationship rather than the simple power relationship seen in the unreduced count, since the reduction itself involves comparing pairs of index values across the entire range.

Illustration

The raw count grows fastest, since it is entirely unconstrained, while the symmetric and antisymmetric counts grow more slowly, with the antisymmetric count remaining consistently the smallest of the three for any given dimension.

Comparing Growth Rates

Raw Count Grows Fastest

Among the three quantities, the raw entry count n raised to the power r grows fastest as dimension increases, since it carries no reduction at all. For rank two, this is exactly n squared, compared to the roughly half that value produced by either symmetric or antisymmetric reduction.

Symmetric and Antisymmetric Counts Converge for Large Dimension

As the dimension n becomes large, both the symmetric count and the antisymmetric count approach one half of n squared, since the difference between them, which is exactly n, becomes proportionally negligible compared to the overall size of n squared. For small dimensions, however, the difference between the two counts, being exactly n, represents a comparatively larger fraction of the total.

Practical Significance

Choosing a Dimension Affects Feasibility

Because the total entry count of a tensor grows according to a power of the dimension, tensors of high rank defined on spaces of large dimension can have an extremely large number of components, even after accounting for any symmetry reduction. Recognizing the Tensor Component Dimension Dependence of a given rank and symmetry pattern allows the practical scale of a tensor's component data to be anticipated before it is computed or stored.

Dimension Independence of the Symmetry Pattern Itself

While the number of components depends on dimension, whether a tensor exhibits symmetric, antisymmetric, or no relationship at all among its indices does not depend on dimension. A tensor's Tensor Component Symmetry Pattern is determined by its defining relationships, and the dimension of the space only affects how many components result from that already-established pattern.

Relationship to Other Tensor Concepts

Tensor Component Dimension Dependence describes how the outputs of the Tensor Component Total Entry Calculation, the Tensor Component Symmetric Reduction, and the Tensor Component Antisymmetric Reduction all vary as the dimension of the underlying space changes, holding the rank and symmetry pattern fixed. It complements Tensor Component Type Dependence, which instead describes how these same counts vary according to the variance type and grouping of a tensor's indices.