／var／logmarcus chiu

Tensor Product

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Tensor Product (·⊗·, ··)
  • vectors in tensor product space 𝑉⊗𝑊 are sums of 𝑣⊗𝑤 for 𝑣∊𝑉 and 𝑤∊𝑊 that obey:
    • 𝑣⊗𝑤1 + 𝑣⊗𝑤2 = 𝑣⊗(𝑤1 + 𝑤2)
    • 𝑣1⊗𝑤 + 𝑣2⊗𝑤 = (𝑣1 + 𝑣2)⊗𝑤
    • (𝜆𝑣)⊗𝑤 = 𝜆(𝑣⊗𝑤) = 𝑣⊗(𝜆𝑤)
  • the tensor product 𝑉⊗𝑊 of two vector spaces 𝑉 and 𝑊 (over the same field) is a vector space to which is associated a bilinear map 𝑉×𝑊 → 𝑉⊗𝑊 that maps a pair (𝑣,𝑤), 𝑣∈𝑉,𝑤∈𝑊 to an element of 𝑉⊗𝑊 denoted as 𝑣⊗𝑤
  • the tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from 𝑉×𝑊 into another vector space 𝑍 factors uniquely through a linear map 𝑉⊗𝑊 → 𝑍
  • this is rigged so that a linear map 𝑉⊗𝑊→𝑍 is the same as a bilinear map 𝑉⨯𝑊→𝑍:
    • for bilinear map 𝑓: 𝑉⨯𝑊→𝑍
      • 𝑓(𝑣,𝑤1+𝑤2) = 𝑓(𝑣,𝑤1) + 𝑓(𝑣,𝑤2) # The simplest bilinear map ℝ×ℝ→ℝ is 𝑓(𝑥,𝑦) = 𝑥𝑦. Thus 𝑓(𝑥,𝑦+𝑧) = 𝑥(𝑦+𝑧) = (𝑥𝑦)+(𝑥𝑧) = 𝑓(𝑥,𝑦)+𝑓(𝑥,𝑧)
    • for the corresponding linear map 𝑓: 𝑉⊗𝑊→𝑍
      • 𝑓(𝑣⊗(𝑤1+𝑤2)) = 𝑓(𝑣⊗𝑤1 + 𝑣⊗𝑤2)      # The corresponding linear map ℝ⊗ℝ→ℝ is 𝑓(𝑥⊗𝑦) = 𝑥⊗𝑦 = [𝑥]⊗[𝑦] = [𝑥[𝑦]] = [𝑥𝑦] = 𝑥𝑦
      • 𝑓(𝑣⊗(𝑤1+𝑤2)) = 𝑓(𝑣⊗𝑤1) + 𝑓(𝑣⊗𝑤2) # Thus 𝑓(𝑥⊗(𝑦+𝑧)) = 𝑥⊗(𝑦+𝑧) = [𝑥]⊗[𝑦+𝑧] = [𝑥[𝑦+𝑧]] = [𝑥𝑦+𝑥𝑧] = [𝑥𝑦] + [𝑥𝑧] = 𝑓(𝑥⊗𝑦)+𝑓(𝑥⊗𝑧)
  • an element of the form 𝑣⊗𝑤 is called the tensor product of 𝑣 and 𝑤
  • an element of 𝑉⊗𝑊 is a tensor
  • the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor
    • the elementary tensors span 𝑉⊗𝑊 in the sense that every element of 𝑉⊗𝑊 is a sum of elementary tensors
  • if bases are given for 𝑉 and 𝑊, a basis of 𝑉⊗𝑊 is formed by all tensor products of a basis element of 𝑉 and a basis element of 𝑊
  • is a generalization of the outer product

Tensor Product - Definition From Bases

Let 𝑉 and 𝑊 be two vector spaces over a field 𝐹, with respective bases 𝐵𝑉 and 𝐵𝑊.

The tensor product of two vectors 𝑣∊𝑉 and 𝑤∊𝑊 is defined from their decomposition on the bases. More precisely, if:

are vectors decomposed on their respective bases, then the tensor product of 𝑣 and 𝑤 is:

  • \begin{align} v \otimes w &=\left(\sum_{b\in B_V} v_b\,b\right) \otimes \left(\sum_{c\in B_W} w_c\,c\right)\\ &=\sum_{b\in B_V}\sum_{c\in B_W} v_b w_c\, b \otimes c \end{align}

where:

  • the coordinate vector of 𝑣⊗𝑤 is the Kronecker product of the column-vectors and/or row-vectors of 𝑣 and 𝑤, which results in a block matrix (preferred representation)
  • the coordinate vector of 𝑣⊗𝑤 is the outer product of the coordinate vectors of 𝑣 and 𝑤, which results in a matrix

It is straightforward to verify that the map (𝑣,𝑤) ↦ 𝑣⊗𝑤 is a bilinear map from 𝑉×𝑊 to 𝑉⊗𝑊.

Tensor Product - Dimension

  • dim(𝑉⊗𝑊) = dim(𝑉) * dim(𝑊)

Tensor Product - Examples

Description

Tensor Product

Kronecker Product

The tensor product of a vector 𝑣∊𝑉 and a covector 𝛼∊𝑉* is a linear map denoted as 𝑣⊗𝛼

The tensor product of a vector space 𝑉 and its dual vector space 𝑉* is a tensor product space denoted as 𝑉⊗𝑉* where each element in that space is a linear map

The tensor product of two covectors 𝛼∊𝑉* and 𝛽∊𝑊* is a bilinear form denoted as 𝛼⊗𝛽

The tensor product of two dual vector spaces 𝑉* and 𝑊* is a tensor product space denoted as 𝑉*⊗𝑊* where each element in that space is a bilinear form

The tensor product of two vectors 𝑣∊𝑉 and 𝑤∊𝑊 is a bilinear map denoted as 𝑣⊗𝑤

The tensor product of two vector spaces 𝑉 and 𝑊 is a tensor product space denoted as 𝑉⊗𝑊 where each element in that space is a bilinear map

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