PureTensor

pure tensor noun, mathematics

Let V and W be vector spaces over a field K.

V,  W   vector spaces over KV,\; W \;\text{ vector spaces over } K

The tensor product V ⊗ W is a vector space equipped with a bilinear map that is universal: for every vector space U over K, every bilinear map V × W → U factors uniquely through it.

⊗:V×W⟶V⊗W,\otimes : V \times W \longrightarrow V \otimes W,(v,w)⟼v⊗w(v, w) \longmapsto v \otimes w
∀ U a vector space over K,\forall\, U \text{ a vector space over } K,∀ β:V×W→U bilinear\forall\, \beta : V \times W \to U \text{ bilinear}∃!  β~:V⊗W→U linear with β=β~∘⊗\exists!\; \tilde{\beta} : V \otimes W \to U \text{ linear with } \beta = \tilde{\beta} \circ \otimes

Bilinearity means the product distributes over addition in each slot and scalars pass through:

(v1+v2)⊗w=v1⊗w+v2⊗w,(v_1 + v_2) \otimes w = v_1 \otimes w + v_2 \otimes w,v⊗(w1+w2)=v⊗w1+v⊗w2v \otimes (w_1 + w_2) = v \otimes w_1 + v \otimes w_2
(λv)⊗w=v⊗(λw)=λ (v⊗w),(\lambda v) \otimes w = v \otimes (\lambda w) = \lambda\,(v \otimes w),λ∈K\lambda \in K

Definition. A tensor t ∈ V ⊗ W is pure (also called simple, elementary, or decomposable) if it is the product of a single vector from each factor. The zero tensor is pure, since 0 = 0 ⊗ w.

t=v⊗wfor some v∈V,  w∈Wt = v \otimes w \quad \text{for some } v \in V,\; w \in W

Every tensor is a finite sum of pure tensors. The least number of terms needed is its tensor rank (the zero tensor is the empty sum, of rank 0), and the nonzero pure tensors are exactly the tensors of rank one.

t=∑i=1rvi⊗wi,t = \sum_{i=1}^{r} v_i \otimes w_i,rank⁡(t)=min⁡{ r≥0  :  t=∑i=1rvi⊗wi}\operatorname{rank}(t) = \min\Big\{\, r \ge 0 \;:\; t = \sum_{i=1}^{r} v_i \otimes w_i \Big\}
t≠0:t pure  ⟺  rank⁡(t)=1t \neq 0 :\quad t \text{ pure} \iff \operatorname{rank}(t) = 1

In bases (eᵢ) of V and (fⱼ) of W the products eᵢ ⊗ fⱼ form a basis of V ⊗ W. A pure tensor has an outer-product coefficient array, and a tensor is pure exactly when its coefficient array has matrix rank at most one.

v=∑iviei,v = \sum_i v^{i} e_i,w=∑jwjfj,w = \sum_j w^{j} f_j,v⊗w=∑i,jviwj ei⊗fjv \otimes w = \sum_{i,j} v^{i} w^{j} \, e_i \otimes f_j
t=∑i,jTij ei⊗fj is puret = \sum_{i,j} T^{ij} \, e_i \otimes f_j \text{ is pure}  ⟺  rank⁡[(Tij)i,j]≤1\iff \operatorname{rank}\big[(T^{ij})_{i,j}\big] \le 1

Not every tensor need be pure. With e₁, e₂ linearly independent in V and f₁, f₂ linearly independent in W, the following tensor has rank two:

e1⊗f1+e2⊗f2   is not puree_1 \otimes f_1 + e_2 \otimes f_2 \;\text{ is not pure}

With more factors the definition is the same: for vector spaces V₁, …, Vₙ over K, a pure tensor in V₁ ⊗ ⋯ ⊗ Vₙ is one of the form

v1⊗v2⊗⋯⊗vn  ∈  V1⊗V2⊗⋯⊗Vn,v_1 \otimes v_2 \otimes \cdots \otimes v_n \;\in\; V_1 \otimes V_2 \otimes \cdots \otimes V_n,vk∈Vkv_k \in V_k
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