7.4 Symmetric and alternating tensors

Symmetric and alternating tensors are crucial concepts in multilinear algebra. They represent different ways tensors behave under index permutations, with symmetric tensors remaining unchanged and alternating tensors changing sign for odd permutations.

These special tensor types have important applications in physics, geometry, and machine learning. Symmetric tensors model stress in materials, while alternating tensors appear as differential forms in geometry. Understanding their properties and constructions is key to mastering tensor algebra.

Symmetric vs Alternating Tensors

Definitions and Properties

Top images from around the web for Definitions and Properties

• 

  Data Science — WRD R&D Documentation View original

  Is this image relevant?

• 

  Levi-Civita symbol - Knowino View original

  Is this image relevant?

• 

  Tensor – AnthroWiki View original

  Is this image relevant?

• 

  Data Science — WRD R&D Documentation View original

  Is this image relevant?

• 

  Levi-Civita symbol - Knowino View original

  Is this image relevant?

1 of 3

Top images from around the web for Definitions and Properties

• 

  Data Science — WRD R&D Documentation View original

  Is this image relevant?

• 

  Levi-Civita symbol - Knowino View original

  Is this image relevant?

• 

  Tensor – AnthroWiki View original

  Is this image relevant?

• 

  Data Science — WRD R&D Documentation View original

  Is this image relevant?

• 

  Levi-Civita symbol - Knowino View original

  Is this image relevant?

1 of 3

• Symmetric tensors remain unchanged under any permutation of indices
• Alternating tensors change sign under odd permutations of indices
• Symmetrization operator Sym constructs symmetric tensors from general tensors
• Alternation operator Alt constructs alternating tensors from general tensors
• Symmetric tensors form subspace S^k(V) of tensor space T^k(V)
• Alternating tensors form subspace Λ^k(V) of tensor space T^k(V)
• Trace of symmetric tensor invariant under cyclic permutations of indices
• Alternating tensors of order k > dim(V) always zero due to antisymmetry property
• Determinant expressed as alternating multilinear form connects alternating tensors to determinants
  • Example: For a 2x2 matrix A = [[a, b], [c, d]], det(A) = ad - bc is an alternating multilinear form

Applications and Examples

• Symmetric tensors used in physics to represent stress and strain in materials
  • Example: Stress tensor σᵢⱼ in continuum mechanics symmetric under index exchange (σᵢⱼ = σⱼᵢ)
• Alternating tensors appear in differential geometry as differential forms
  • Example: The electromagnetic field tensor Fμν in relativity is an alternating tensor
• Symmetric tensors used in machine learning for covariance matrices
  • Example: Covariance matrix Σ in multivariate statistics symmetric (Σᵢⱼ = Σⱼᵢ)
• Alternating tensors used in exterior calculus for integration on manifolds
  • Example: Volume form on a 3D manifold represented by alternating 3-tensor εᵢⱼₖ

Constructing Tensor Algebras

Symmetric Algebra

• Symmetric algebra S(V) direct sum of all symmetric tensor spaces S^k(V) for k ≥ 0
• S(V) equipped with multiplication operation
• S(V) isomorphic to polynomial algebra on basis of V
  • Example: For V = span{x, y}, S(V) ≅ ℝ[x, y], the polynomial ring in two variables
• S(V) graded algebra with grading given by tensor order
• Universal property allows unique extension of linear maps to algebra homomorphisms
  • Example: Linear map f: V → A (A commutative algebra) extends uniquely to F: S(V) → A

Exterior Algebra

• Exterior algebra Λ(V) direct sum of all alternating tensor spaces Λ^k(V) for k ≥ 0
• Λ(V) equipped with wedge product operation
• Λ(V) quotient of tensor algebra T(V) by ideal generated by v ⊗ v for v ∈ V
• Λ(V) graded algebra with grading given by tensor order
• Exterior algebra used in differential geometry to define differential forms
  • Example: Λ(ℝ³) used to represent scalar, vector, and pseudovector fields in 3D space

Dimensions of Tensor Spaces

Symmetric Tensor Space Dimensions

• Dimension of S^k(V) given by binomial coefficient ${n+k-1 \choose k}$, where n = dim(V)
• Proof uses correspondence between symmetric tensors and homogeneous polynomials
  • Example: For V = ℝ³, dim(S²(V)) = ${5 \choose 2}$ = 10, corresponding to x², y², z², xy, xz, yz, x²y, x²z, y²z, xyz
• Generating function for dimension sequence of S(V) $(1-t)^{-n}$
  • Example: For V = ℝ², generating function $(1-t)^{-2} = 1 + 2t + 3t² + 4t³ + ...$

Alternating Tensor Space Dimensions

• Dimension of Λ^k(V) given by binomial coefficient ${n \choose k}$, where n = dim(V)
• Proof utilizes exterior power construction and properties of alternating multilinear forms
  • Example: For V = ℝ⁴, dim(Λ²(V)) = ${4 \choose 2}$ = 6, corresponding to dx∧dy, dx∧dz, dx∧dw, dy∧dz, dy∧dw, dz∧dw
• Generating function for dimension sequence of Λ(V) $(1+t)^n$
  • Example: For V = ℝ³, generating function $(1+t)^3 = 1 + 3t + 3t² + t³$

Comparative Analysis

• dim(S^k(V)) > dim(Λ^k(V)) for k > 1 and n > 1
  • Example: For V = ℝ⁴, dim(S²(V)) = 10 while dim(Λ²(V)) = 6
• Dimension formulas used to derive combinatorial identities
  • Example: ${n \choose k} + {n \choose k-1} = {n+1 \choose k}$ derived from exterior algebra dimensions

Wedge Product for Alternating Tensors

Properties and Operations

• Wedge product ∧ associative, bilinear operation producing alternating tensors
• For vectors v and w, v ∧ w = -w ∧ v captures antisymmetry property
• Wedge product of k vectors zero if and only if vectors linearly dependent
  • Example: In ℝ³, (1,0,0) ∧ (0,1,0) ∧ (1,1,0) = 0 as vectors are linearly dependent
• Wedge product constructs basis for Λ^k(V) from basis of V
  • Example: For V = span{e₁, e₂, e₃}, basis of Λ²(V) e₁∧e₂, e₁∧e₃, e₂∧e₃

Applications in Linear Algebra and Geometry

• Determinant of matrix A computed as wedge product of column vectors
  • Example: For 2x2 matrix A = [[a,b],[c,d]], det(A) = (a,c) ∧ (b,d) = ad - bc
• Exterior derivative in differential geometry defined using wedge product
  • Example: d(f dx + g dy) = (∂f/∂y - ∂g/∂x) dx ∧ dy for 2-form in ℝ²
• Grassmann algebra generalizes vector algebra using exterior algebra structure
  • Example: Plücker coordinates in projective geometry represented using wedge products