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
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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 k ) {n+k-1 \choose k} ( k n + k − 1 ) , where n = dim(V)
Proof uses correspondence between symmetric tensors and homogeneous polynomials
Example: For V = ℝ³, dim(S²(V)) = ( 5 2 ) {5 \choose 2} ( 2 5 ) = 10, corresponding to x², y², z², xy, xz, yz, x²y, x²z, y²z, xyz
Generating function for dimension sequence of S(V) ( 1 − t ) − n (1-t)^{-n} ( 1 − t ) − n
Example: For V = ℝ², generating function ( 1 − t ) − 2 = 1 + 2 t + 3 t 2 + 4 t 3 + . . . (1-t)^{-2} = 1 + 2t + 3t² + 4t³ + ... ( 1 − t ) − 2 = 1 + 2 t + 3 t 2 + 4 t 3 + ...
Alternating Tensor Space Dimensions
Dimension of Λ^k(V) given by binomial coefficient ( n k ) {n \choose k} ( k n ) , where n = dim(V)
Proof utilizes exterior power construction and properties of alternating multilinear forms
Example: For V = ℝ⁴, dim(Λ²(V)) = ( 4 2 ) {4 \choose 2} ( 2 4 ) = 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 (1+t)^n ( 1 + t ) n
Example: For V = ℝ³, generating function ( 1 + t ) 3 = 1 + 3 t + 3 t 2 + t 3 (1+t)^3 = 1 + 3t + 3t² + t³ ( 1 + t ) 3 = 1 + 3 t + 3 t 2 + t 3
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 k ) + ( n k − 1 ) = ( n + 1 k ) {n \choose k} + {n \choose k-1} = {n+1 \choose k} ( k n ) + ( k − 1 n ) = ( k n + 1 ) 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