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Math Notes

University Mathematics

ConceptComplete

Properties of Tensor Products

Tensor products satisfy algebraic properties that make them a powerful tool for constructing new vector spaces. Associativity, commutativity, and distributivity govern tensor algebra.

TheoremAlgebraic Properties

For vector spaces U,V,WU, V, W over F\mathbb{F}:

  1. Associativity: (U⊗V)⊗W≅U⊗(V⊗W)(U \otimes V) \otimes W \cong U \otimes (V \otimes W)
  2. Commutativity: V⊗W≅W⊗VV \otimes W \cong W \otimes V
  3. Distributivity: U⊗(V⊕W)≅(U⊗V)⊕(U⊗W)U \otimes (V \oplus W) \cong (U \otimes V) \oplus (U \otimes W)
  4. Identity: V⊗F≅VV \otimes \mathbb{F} \cong V

These isomorphisms are canonical (natural, not depending on choices).

DefinitionTensor Product of Linear Maps

If T:V→V′T: V \to V' and S:W→W′S: W \to W' are linear maps, define: T⊗S:V⊗W→V′⊗W′T \otimes S: V \otimes W \to V' \otimes W' (v⊗w)↦T(v)⊗S(w)(\mathbf{v} \otimes \mathbf{w}) \mapsto T(\mathbf{v}) \otimes S(\mathbf{w})

Extend linearly to all of V⊗WV \otimes W. This makes tensor product a functor.

In matrices: if TT has matrix AA and SS has matrix BB, then T⊗ST \otimes S has matrix A⊗BA \otimes B (Kronecker product).

ExampleKronecker Product

For A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} and B=[efgh]B = \begin{bmatrix} e & f \\ g & h \end{bmatrix}:

A⊗B=[aBbBcBdB]=[aeafbebfagahbgbhcecfdedfcgchdgdh]A \otimes B = \begin{bmatrix} aB & bB \\ cB & dB \end{bmatrix} = \begin{bmatrix} ae & af & be & bf \\ ag & ah & bg & bh \\ ce & cf & de & df \\ cg & ch & dg & dh \end{bmatrix}

This 4×44 \times 4 matrix represents the tensor product linear map.

TheoremTensor Products and Direct Sums

The tensor product distributes over direct sums: V⊗(W1⊕W2)≅(V⊗W1)⊕(V⊗W2)V \otimes (W_1 \oplus W_2) \cong (V \otimes W_1) \oplus (V \otimes W_2)

This allows decomposing tensor products into simpler pieces.

DefinitionMultilinear Maps

A map ϕ:V1×V2×⋯×Vk→W\phi: V_1 \times V_2 \times \cdots \times V_k \to W is multilinear if it's linear in each argument separately.

The tensor product generalizes to V1⊗V2⊗⋯⊗VkV_1 \otimes V_2 \otimes \cdots \otimes V_k with universal property: multilinear maps from V1×⋯×VkV_1 \times \cdots \times V_k correspond bijectively to linear maps from V1⊗⋯⊗VkV_1 \otimes \cdots \otimes V_k.

Remark

These properties echo those of multiplication in algebra: associativity and commutativity make tensor products behave like "multiplication of vector spaces." The dimension formula dim⁡(V⊗W)=dim⁡(V)⋅dim⁡(W)\dim(V \otimes W) = \dim(V) \cdot \dim(W) reinforces this analogy. Tensor products appear throughout modern mathematics: in differential geometry (tensor fields), algebraic topology (homology), and quantum mechanics (entangled states).