6.4.3 Tensor Degree Component Index Relation

Tensor Degree Component Index Relation explains how indices in tensor components relate to their degree, defining structure in multilinear algebra.

Tensor Degree Component Index Relation is the statement of how many integer indices are needed to address a component of an element of the tensor algebra T(V) once a basis is fixed: a homogeneous element of degree n has components addressed by exactly n indices, forming a single n-dimensional array, while a general, inhomogeneous element has no single such array at all, requiring instead one separate array of a different dimensionality for each degree appearing in its decomposition. This relation extends the component-indexing facts already established for order classification to the graded setting of T(V), making explicit exactly when a single flat array suffices and when it does not.

Component Indexing for a Homogeneous Element

One Index per Degree Unit

For x homogeneous of degree n, fixing a basis {e_i} of V lets every component of x be written x_{i₁...i_n}, addressed by an n-tuple of integers each ranging over the dimension of V:

x_{i_{1} \dots i_{n}}, each i_{m} ranging from 1 to dim(V)

This is exactly the component-index count already established for an order-n tensor, restated here for the case where the tensor happens to be a homogeneous element of T(V).

A Single n-Dimensional Array Suffices

Because every component of a homogeneous degree-n element is addressed by the same number of indices, n, the entire element can be stored in a single, uniform n-dimensional array with dim(V)^n entries, exactly as would be done for any order-n tensor considered on its own.

Component Indexing for a General Element

No Single Array Can Hold a Mixed-Degree Element

A general element x = x₀ + x₁ + x₂ has a degree-0 part addressable by zero indices (a single number), a degree-1 part addressable by one index (a list), and a degree-2 part addressable by two indices (a two-dimensional array); no single array shape can accommodate all three parts simultaneously, since they require different numbers of indices.

A Collection of Arrays, One per Degree

Representing a general element of T(V) therefore requires a collection of arrays, one for each degree present in its decomposition, each with its own appropriate number of dimensions:

x \leftrightarrow {x_{0}, ({x 1}_{i}), ({x 2}_{ij}), \dots}

with the understanding that only finitely many of these arrays are nonzero for any given element, since a general element of T(V) is by definition a finite sum of homogeneous parts.

Diagram of Array Shapes by Degree

Total Component Count Across Degrees

Summing Component Counts by Degree

For an element with nonzero parts up to some maximum degree N, the total number of stored components across all its arrays is the sum of the component counts of each part:

total components = \sum_{n = 0}^{N} \dim (V)^{n}

a sum that mixes contributions of different array shapes rather than a single product formula, in contrast to the pure-power formula that applies to one homogeneous part alone.

Why the Sum, Not a Product, Governs the General Case

The product formula dim(V)^n correctly counts the components of one homogeneous degree-n part; because the parts of different degrees are stored in separate, differently shaped arrays rather than combined into one larger array, the total across all parts is obtained by adding these separate counts, not by any single exponentiation.

Why This Relation Matters

Determining Storage Requirements for General Algebra Elements

Any computational representation of a general element of T(V), as opposed to a single homogeneous tensor, must anticipate that it will need to store several arrays of different dimensionality rather than one, a direct practical consequence of this relation for anyone implementing tensor-algebra arithmetic.

Reinforcing the Boundary Between Homogeneous and General Elements

The sharp contrast between "one array suffices" for a homogeneous element and "a separate array per degree is required" for a general element reinforces, from the concrete angle of component storage, the same boundary already identified between order classification (built for homogeneous, single-order tensors) and degree classification (built to accommodate the fully general, possibly mixed-degree elements of T(V)).