15.13.5 Tensor Symmetric Rank Decomposition Relation

Tensor Symmetric Rank Decomposition Relation explains how symmetric tensors decompose into sums of rank-one components, linking algebraic structures and geometric insights.

Tensor Symmetric Rank Decomposition Relation is the explicit algebraic identity that a proposed set of scalars and vectors must satisfy in order to constitute a genuine decomposition of a given symmetric tensor into pure power terms, expressed both at the level of raw tensor components and, equivalently, at the level of the associated homogeneous polynomial, providing the precise equation that any candidate decomposition is checked against. Where the pure power term describes the shape of a single summand and the minimality condition describes what makes a term count qualify as the rank, the decomposition relation is the equation itself, the concrete statement of equality that links a chosen collection of terms to the specific tensor they are claimed to represent.

Stating this relation explicitly, in both the tensor and polynomial forms, is what makes the abstract idea of decomposing a symmetric tensor into pure powers into a concretely verifiable and constructible procedure, since any proposed decomposition can be checked by direct substitution into this relation, and any search for a decomposition can be organized as a search for scalars and vectors satisfying it.

The Relation in Tensor Component Form

Componentwise Equality

For a rank-n symmetric tensor T with components T_{i1...in}, a decomposition into r pure power terms, given by scalars c_1 through c_r and vectors v_1 through v_r, must satisfy the decomposition relation at every index tuple:

T_{i_{1} \dots i_{n}} = \sum_{k = 1}^{r} c_{k} v_{k}^{i_{1}} \dots v_{k}^{i_{n}}

where v_k^{ij} denotes the ij-th coordinate of the vector v_k, so each term in the sum contributes the product of the relevant coordinates of a single vector, weighted by its scalar coefficient.

Reduction to Independent Components

Because both sides of this relation automatically satisfy the symmetric equality constraint, the relation need only be verified at the independent, canonical index tuples rather than at every one of the d^n possible tuples, reducing the number of scalar equations that must hold to exactly C(d, n), matching the count of independent components.

The Relation in Polynomial Form

Equality of Associated Homogeneous Polynomials

Under the polynomial analogy, the decomposition relation restates equivalently as an equality between the homogeneous polynomial Q associated with T and a sum of r scalar multiples of perfect n-th powers of linear forms:

Q (x) = \sum_{k = 1}^{r} c_{k} {(ℓ_{k} (x))}^{n}

where ℓ_k is the linear form associated with v_k; this polynomial form of the relation is the classical statement studied in the Waring problem for polynomials, expressing a given homogeneous polynomial as a sum of powers of linear forms.

Coefficient Matching

Expanding the right-hand side using the multinomial theorem and comparing coefficients of each monomial against the corresponding coefficients of Q produces exactly the same system of C(d, n) scalar equations obtained directly from the tensor component form, confirming that the two statements of the relation, tensor and polynomial, carry identical information.

Solving the Relation

A Nonlinear System in the Unknown Vectors

Although the relation is linear in the scalars c_k once the vectors v_k are fixed, it is generally nonlinear in the vectors themselves, since each term contributes a product of n coordinates of v_k; finding a decomposition therefore typically requires solving a system of polynomial equations in the unknown vector components, a substantially harder problem than solving a linear system.

The Role of r in Solvability

For a fixed target tensor T, the decomposition relation becomes solvable once r is large enough, since increasing r increases the number of free parameters, namely the components of each v_k together with each c_k, relative to the fixed number of equations, C(d, n); the minimal r for which the relation first becomes solvable is exactly the symmetric rank of T, tying the decomposition relation's solvability directly back to the rank structure's central invariant.

Uniqueness Considerations

Generic Uniqueness for Minimal Decompositions

For many combinations of rank n and dimension d, a minimal decomposition satisfying the relation is, once the individual pure power term ambiguities of scalar and sign are accounted for, unique for a generic tensor T, a fact underlying results on the identifiability of symmetric tensor decompositions; this uniqueness does not hold universally, and certain special combinations of n and d admit tensors with more than one essentially distinct minimal decomposition satisfying the relation.

Verification as a Direct Consequence of the Relation

Regardless of how a candidate decomposition was found, whether by direct construction or by an iterative numerical method, its correctness is always verifiable by direct substitution into the decomposition relation, checking equality against every independent component of T, providing a definitive, self-contained criterion for confirming that a proposed set of pure power terms genuinely reconstructs the target tensor.