5.7.5 Tensor Simple Element Rank Relation

Tensor Simple Element Rank Relation is the precise correspondence between simple (elementary, decomposable) tensors and the value one for tensor rank: a tensor is simple exactly when its tensor rank is at most one, and this equivalence is the hinge connecting the qualitative notion of simplicity to the quantitative notion of rank used throughout tensor decomposition theory.

Statement of the Relation

Let t be an element of a tensor product T = V1 ⊗ V2 ⊗ ⋯ ⊗ Vn over a field F. The tensor rank of t, denoted rank(t), is defined as the smallest number r for which t can be written as a sum of r simple tensors,

The rank relation asserts:

with the convention that the zero tensor has rank zero, and every nonzero simple tensor has rank exactly one.

Proof Sketch of the Two Directions

The equivalence splits into two directions, each straightforward from the definitions but worth separating because they justify different practical uses of the relation.

Simple Implies Rank at Most One

If t is simple, it is by definition a single tensor product of one factor per space, which is exactly the r = 1 case of the sum-of-simple-tensors decomposition defining rank. Hence rank(t) ≤ 1 immediately.

Rank at Most One Implies Simple

Conversely, if rank(t) ≤ 1, then either t is the zero tensor, which is conventionally excluded from being called simple, or t equals a single term v1 ⊗ ⋯ ⊗ vn in a minimal decomposition, which is precisely the defining condition for t to be simple. No additional argument is required beyond unpacking the definition of rank at r = 1.

Rank as a Distance from Simplicity

Once the base case is settled, rank for larger values measures how many simple tensors are needed to build up a general element, giving rank a natural interpretation as a graded measure of complexity anchored at simple tensors.

Rank Two and Beyond

A tensor with rank exactly two is, by definition, not simple, since if it were expressible as a single simple term, its rank would be one rather than two — a direct consequence of the rank relation combined with the minimality built into the definition of rank. More generally, no tensor of rank r > 1 is simple, so the simple element rank relation partitions all nonzero tensors into the single class of rank-one (simple) tensors and the remaining classes of rank r ≥ 2 tensors, of which there are typically infinitely many rank strata for large enough factor dimensions.

Rank as a Stratification

The rank relation is the base level of a broader stratification of the tensor product space into sets of tensors of rank exactly r, for r = 0, 1, 2, and so on up to the maximal possible rank. The simple tensors — rank exactly one — form the smallest nontrivial stratum in this stratification, sitting as a thin, nonlinear subset just outside the origin (rank zero).

Matrix Case as the Model Instance

The rank relation is most transparent, and most classically understood, in the two-factor case, where tensors correspond to matrices and tensor rank coincides with ordinary matrix rank.

Rank-One Matrices

A matrix M has rank at most one exactly when it can be written as an outer product M = a bᵀ of two vectors, which is precisely the simple product form for a two-factor tensor. The equivalence between "rank at most one" and "simple" is therefore the same statement, in the matrix setting, as the familiar fact that rank-one matrices are exactly the outer products of vector pairs, extended verbatim to n factors via the same rank relation.

Divergence at Higher Order

For three or more factors, the analogy between matrix rank and tensor rank becomes subtler, since higher-order tensor rank can exceed the dimensions of the factor spaces in ways matrix rank cannot, and computing rank exactly becomes computationally hard in general. Nonetheless, the base case of the rank relation — rank one corresponds to simple — remains completely elementary and true without qualification at any order n, since it follows directly from the definitions rather than from any deeper structural theorem.

Consequences for Recognition and Decomposition

The rank relation supplies the theoretical justification for treating simple-element recognition and low-rank decomposition as two views of the same underlying question.

Recognition as a Rank Test

Because simplicity is equivalent to rank at most one, any algorithm that computes or bounds tensor rank automatically doubles as a recognition procedure for simple tensors, by checking whether the computed or bounded rank equals one. This is why simple element recognition, in the two-factor case, reduces cleanly to a matrix rank test.

Best Approximation Anchored at Rank One

The best rank-one approximation problem, central to many decomposition algorithms, is directly framed by the rank relation as the search for the simple tensor closest to a given tensor, since a rank-one tensor is by the relation exactly a simple tensor. Any theoretical or algorithmic result about approximating tensors by rank-one terms is, through this relation, equally a statement about approximating tensors by simple tensors.

Illustrative Diagram

The shaded block marks the rank-one stratum, which the rank relation identifies exactly with the set of simple tensors, distinguishing it from the zero tensor on one side and higher-rank tensors on the other.