15.8.1 Tensor Symmetric Product Factor Selection

Tensor Symmetric Product Factor Selection is the question of which tensors, or which combinations of vectors, are chosen as the inputs to a symmetric product operation, together with the consequences that different choices of factors have for the rank, the independent components, and the redundancy of the resulting symmetric tensor. Because the symmetric product is commutative and associative, the identity of the factors, not their order or grouping, is what determines the output uniquely, making factor selection the primary design decision when building a symmetric tensor from simpler pieces via this operation.

Factor selection matters practically because not every symmetric tensor of a given rank arises as a symmetric product of lower-rank factors in an obviously simple way; some symmetric tensors are sums of several such products rather than a single product, and understanding how factor selection shapes the resulting tensor clarifies both which tensors are easy to construct directly and which require combining multiple symmetric products.

Selecting Vector Factors

Repeated Versus Distinct Vectors

Choosing the same vector v for all n factor slots in an n-fold symmetric product yields, up to the interpretive correspondence with the associated homogeneous polynomial, a tensor closely tied to the single monomial v raised to the n-th power in the quadratic-or-higher relation; choosing n distinct vectors u_1 through u_n instead produces a symmetric tensor whose associated homogeneous polynomial is the fully expanded product of n distinct linear factors, containing cross terms among all the chosen vectors.

Effect on Independent Components

Regardless of whether the selected vector factors are repeated or distinct, the resulting symmetric product always lies in the same result space, the rank-n symmetric subspace, and is subject to the same independent component count; factor selection changes which specific tensor within that subspace is produced, not the dimension of the subspace itself.

Selecting Tensor Factors of Mixed Rank

Combining a Symmetric Tensor With a Vector

A common factor selection is to combine an existing rank-p symmetric tensor T with a single additional vector v, producing a rank-(p+1) symmetric tensor T odot v; this selection is often used to build up higher-rank symmetric tensors incrementally, one vector at a time, rather than assembling all factors simultaneously.

Combining Two Nontrivial Symmetric Tensors

Selecting two symmetric tensors T and R of rank p and q greater than one each, rather than building purely from vectors, produces a rank-(p+q) symmetric tensor whose structure reflects the internal symmetry of both T and R simultaneously; this selection is used when the pieces being combined already carry meaningful symmetric structure of their own, rather than starting from the most elementary rank-one building blocks.

Selection and the Spanning Question

Symmetric Products of Vectors Span the Result Space

Every totally symmetric tensor of rank n can be written as a finite sum of symmetric products of n vectors, meaning the collection of all possible vector-factor selections, together with linear combination, is sufficient to reach every tensor in the rank-n symmetric subspace; no symmetric tensor lies outside the span of such products.

A Single Selection Rarely Suffices

Despite this spanning property, a single symmetric product of n vectors, chosen once, generally represents only a very specific tensor within the rank-n symmetric subspace, since the associated homogeneous polynomial of such a single product factors completely into n linear pieces; a generic element of the symmetric subspace corresponds to a homogeneous polynomial that does not factor this way, and reaching it requires selecting and summing several distinct symmetric products rather than relying on one factor selection alone.

Factor Selection and the Symmetric Product Rank Formula

Degree Addition Constrains but Does Not Determine the Selection

The degree addition rule fixes the rank of the output once the ranks of the selected factors are known, but it places no constraint on which specific tensors of those ranks are chosen; factor selection is therefore the remaining degree of freedom left open after the rank of the desired output has been decided, and different selections consistent with the same target rank generally produce different resulting tensors.

Practical Considerations in Choosing Factors

When constructing a symmetric tensor with prescribed values on its independent components, factor selection is typically approached by matching the associated homogeneous polynomial: choosing vector or tensor factors whose symmetric product's associated polynomial reproduces the target polynomial, using the correspondence between symmetric tensors and homogeneous polynomials established through the multilinear polarization context, guides the selection toward factors that achieve a specific desired result.