Symmetric and Alternating Tensor Products Symmetric Tensor Product: • The symmetric tensor product of vector spaces captures commutativity in tensors. • Formally, given V ⊗ V , the symmetric tensor product is defined as: v⊗w+w⊗v for all v, w ∈ V. • The resulting space is denoted S 2 (V ), with dimension: dim(V ) + 1 2 dim(S (V )) = . 2 • Example: For V = R3 , dim(S 2 (V )) = 3+1 2 = 6. Alternating Tensor Product: • The alternating tensor product encodes antisymmetry in tensors. • Formally, given V ⊗ V , the alternating tensor product satisfies: v⊗w−w⊗v for all v, w ∈ V. • The resulting space is denoted ∧2 (V ), with dimension: dim(V ) 2 dim(∧ (V )) = . 2 • Example: For V = R4 , dim(∧2 (V )) = 4 2 = 6. Applications to Multilinear Algebra Tensor Contraction: • Tensor contraction reduces the order of a tensor by pairing indices. • Example: Let T ∈ V ⊗W ⊗U . Contracting between V and U produces an element in W : X Contract(T ) = Ti,j,k δ ik . i • Tensor contraction is widely used in physics, particularly in general relativity, where the Einstein summation convention simplifies operations. Decomposition of Tensors: • Any tensor in V ⊗ W can be written as a sum of simple tensors: X Examples and Exercises Example 1: Symmetric Tensor Product of Basis Elements • Let V = R2 , with basis elements {e1 , e2 }. • The symmetric tensor product S 2 (V ) has basis: {e1 ⊗ e1 , e1 ⊗ e2 + e2 ⊗ e1 , e2 ⊗ e2 }. • Therefore, dim(S 2 (V )) = 3. Example 2: Alternating Tensor Product of Basis Elements • Let V = R3 , with basis elements {e1 , e2 , e3 }. • The alternating tensor product ∧2 (V ) has basis: {e1 ∧ e2 , e1 ∧ e3 , e2 ∧ e3 }. • Therefore, dim(∧2 (V )) = 3. Example 3: Tensor Contraction in Matrix Representation • Let A ∈ R2×3 and B ∈ R3×2 . Compute the contraction tr(A ⊗ B). • The tensor product A ⊗ B is a matrix in R6×6 : Contract(A ⊗ B) = 2 X 3 X Aij Bji . i=1 j=1 Exercise 1: Symmetric Decomposition • Decompose the symmetric tensor product of V = R2 into its basis elements. • Compute the dimension of S 3 (V ) for V = R4 . Exercise 2: Alternating Tensor Applications • Prove that any alternating tensor on R2 is degenerate. • Find the basis for ∧3 (R4 ) and compute its dimension. 2 Visualizations and Diagrams Diagram 1: Symmetric Tensor Product Basis e1 ⊗ e1 e1 ⊗ e2 + e2 ⊗ e1 e2 ⊗ e2 Diagram 2: Alternating Tensor Product Basis e1 ∧ e2 e1 ∧ e3 e2 ∧ e3 Diagram 3: Higher-Order Tensor Visualization v⊗w⊗u v⊗u⊗w 3 w⊗v⊗u