15.13.2 Tensor Symmetric Rank Pure Power Term

Tensor Symmetric Rank Pure Power Term is a single summand of the form c times v^{odot n}, a scalar multiple of the n-th symmetric power of one vector v, appearing as one contribution within a decomposition of a general rank-n symmetric tensor into a sum of such terms. Because a pure power term corresponds, under the polynomial analogy, to a scalar multiple of a single perfect n-th power of a linear form, it represents the simplest possible nonzero building block available for assembling an arbitrary symmetric tensor, and understanding its properties is the necessary first step before examining how many such terms are required to express a general symmetric tensor as their sum.

Studying the pure power term in isolation, before addressing decomposition into several such terms, clarifies exactly what structural features distinguish a tensor built from one power term from a generic symmetric tensor, since not every symmetric tensor is itself a single pure power term, and recognizing the difference is essential context for the broader question of symmetric tensor rank.

Structure of a Single Pure Power Term

General Form

A pure power term takes the form:

with c a scalar and v a vector in the underlying vector space; the rank of this tensor, in the ordinary sense of tensor rank inherited from the symmetric power structure, is n, matching the degree of the power.

Associated Polynomial of a Single Term

Under the polynomial analogy, this pure power term corresponds to the polynomial c times ℓ(x)^n, where ℓ is the linear form associated with v, meaning the associated homogeneous polynomial of a pure power term is, up to the scalar multiple c, a perfect n-th power of a single linear expression, the most restrictive possible factorization pattern a degree-n homogeneous polynomial can exhibit.

Identifying Whether a Given Tensor Is a Pure Power Term

The Perfect Power Condition on the Associated Polynomial

A rank-n symmetric tensor T is expressible as a single pure power term precisely when its associated homogeneous polynomial factors as a perfect n-th power of some linear form, a condition that fails for the overwhelming majority of symmetric tensors of rank two or higher, since a generic homogeneous polynomial of degree n does not factor into n identical linear pieces.

Rank-Two Illustration

For rank two, a symmetric tensor corresponds to a pure power term exactly when its associated quadratic form is a perfect square, equivalently when its matrix representation has rank one in the ordinary matrix sense; a generic symmetric matrix has full matrix rank rather than matrix rank one, illustrating concretely why most rank-two symmetric tensors are not single pure power terms.

Non-Uniqueness of the Term Itself

Scalar and Sign Ambiguity

A pure power term c v^{odot n} can be rewritten using a different vector and scalar producing the same tensor, since replacing v by a scalar multiple lambda v and c by c divided by lambda to the n compensates exactly; when n is even, replacing v by its negation and leaving c unchanged also reproduces the same term, since negative one raised to an even power is one, so a single pure power term does not correspond to a unique choice of v and c without an additional normalization convention.

Consequence for Counting Terms in a Decomposition

This non-uniqueness at the level of a single term means that when several pure power terms are summed to build a general symmetric tensor, the specific vectors and scalars chosen are not unique even once the number of terms is fixed, though the minimal number of terms required, addressed separately as the symmetric tensor's rank in the decomposition sense, is typically a well-defined and more robust invariant.

The Pure Power Term as the Atomic Unit of Decomposition

Building Blocks for General Symmetric Tensors

Because every element of Sym^n(V) is a linear combination of symmetric powers of vectors, as established when discussing the spanning property of symmetric powers, the pure power term is exactly the atomic unit from which every symmetric tensor of rank n is assembled, with the total number of such units in a minimal decomposition determining the tensor's rank.

Relation to the Basis Expansion

While the symmetric basis structure expresses a tensor as a combination of a fixed set of basis power terms e^alpha, a general decomposition into pure power terms permits arbitrary vectors, not only basis vectors, as the base of each term, offering a potentially far more efficient representation, using fewer terms than the number of basis elements required by the fixed-basis expansion, a distinction central to the broader study of symmetric tensor rank.