6.23.2 Tensor Component Count by Order

Tensor Component Count by Order is the study of how the total number of scalar components of a tensor grows as its order n = p + q increases while the dimension d of the underlying vector space is held fixed, treating order as the primary variable of interest in the formula N = d^n. Because order enters this formula as an exponent rather than as a base, component count by order is dominated by exponential growth, and this topic isolates the consequences of that exponential behavior for the algebra, storage, and manipulation of tensors as their order increases.

Order as the Exponent

The Formula Restated with Order as the Variable

Fixing the dimension d and letting the order n vary, the component count is

an exponential function of n with fixed base d. Each increase of the order by one multiplies the component count by another factor of d, since one additional index slot is added, and that slot independently ranges over all d basis directions.

The Recursive View

The exponential growth can equivalently be understood recursively: a tensor of order n + 1 is built by attaching one more vector-space or dual-space factor to a tensor of order n, and each of the N(n) existing components is paired with each of the d new basis choices, giving N(n+1) = d · N(n). This recursive relation, together with the base case N(0) = 1 for scalars, generates the same formula N(n) = dⁿ by unfolding the recursion.

Growth Table for Representative Dimensions

Comparing Small Dimensions

Both sequences double or quadruple at each step respectively, but neither ever slows down; unlike polynomial growth in dimension, exponential growth in order has no diminishing rate of increase, and the ratio between successive terms is always exactly d.

Diagram of Order-Driven Exponential Growth

Consequences of Exponential Growth in Order

The Curse of Dimensionality in Multilinear Algebra

Because N(n) grows exponentially in order regardless of how small d is (as long as d ≥ 2), tensors of high order become intractable to store or manipulate explicitly even over comparatively low-dimensional spaces; a type (0, 10) tensor over a 10-dimensional space already requires 10^10 components, illustrating why algorithms working with high-order tensors favor implicit or compressed representations over direct component-wise storage.

Grading of the Full Tensor Algebra by Order

The exponential component count by order is also what gives structure to the full tensor algebra ⊕ₙ (⊗ⁿV), the direct sum over all orders n, since each graded piece ⊗ⁿV has dimension dⁿ; the tensor algebra as a whole is infinite-dimensional precisely because this sequence of dimensions 1, d, d², d³, ... never terminates and never repeats, growing without bound as n increases.

Contrast With Symmetric and Antisymmetric Subspaces

When a tensor is constrained to be totally symmetric or totally antisymmetric in all of its indices, the independent component count by order grows more slowly than dⁿ: the antisymmetric case grows as a binomial coefficient C(d, n), which is zero once n > d, in sharp contrast to the unconstrained exponential growth. This contrast highlights that the plain exponential formula dⁿ describes the largest possible component count for a given order and dimension, realized only when no symmetry constraints reduce the count of independent components.