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Symmetric tensor

In mathematics, a symmetric tensor is a multilinear map or an array of components that remains invariant under the permutation of its indices, meaning that swapping any two indices does not alter its value.^[1]^[2] For a second-rank tensor A, this symmetry condition is expressed as A^{mn} = A^{nm}, distinguishing it from antisymmetric tensors where components change sign under such exchanges.^[1] Symmetric tensors generalize the notion of symmetric matrices, which are second-order cases where the matrix equals its own transpose, to higher-order tensors that arise in multilinear algebra and tensor analysis.^[3] Any tensor can be uniquely decomposed into the sum of a symmetric part and an antisymmetric part, with the symmetric component given by \frac{[1](/page/1)}{2}(T_{ab} + T_{ba}) for rank-two cases, allowing for the isolation of symmetric structures in physical and geometric applications.^[1]^[2] In three dimensions, a symmetric rank-two tensor has six independent components and can be further split into a trace (scalar) part and a traceless part, facilitating representations in contexts like stress tensors in continuum mechanics.^[2] Key properties of symmetric tensors include their ability to be diagonalized in orthonormal bases for rank-two cases, where they possess real eigenvalues and orthogonal eigenvectors, and their decomposition into sums of rank-one symmetric terms like outer products v \otimes \cdots \otimes v for higher ranks, known as the symmetric tensor rank.^[3]^[2] These features underpin applications in fields such as signal processing, blind source identification, and algebraic geometry, where the minimal number of such rank-one terms (symmetric rank) determines complexity and identifiability.^[3] The product of a symmetric tensor and an antisymmetric one always vanishes, highlighting their orthogonal roles in tensor spaces.^[1]

Definition and Basics

Definition

In multilinear algebra, a tensor of type (k,0) is a multilinear map from the Cartesian product of k copies of a vector space V to the underlying field, such as the real or complex numbers.^[4] A tensor T of type (k,0) over a vector space V is symmetric if it is invariant under any permutation of its arguments, meaning T(v_1, \dots, v_k) = T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) for all vectors v_1, \dots, v_k \in V and any permutation \sigma in the symmetric group S_k.^[4] In coordinates with respect to a basis of V, the components T_{i_1 \dots i_k} of a symmetric tensor satisfy T_{i_1 \dots i_k} = T_{j_1 \dots j_k} whenever (j_1, \dots, j_k) is any permutation of (i_1, \dots, i_k).^[5] For mixed tensors of type (k,l), symmetry can be defined separately with respect to the k contravariant indices or the l covariant indices, requiring invariance under permutations within each group of like indices; full symmetry across all indices is possible but less common unless specified.^[6]

Examples

Symmetric tensors appear in various mathematical and physical contexts, providing concrete illustrations of their defining property of invariance under index permutation. For instance, a rank-0 tensor, which is simply a scalar such as mass or temperature, possesses no indices and is thus trivially symmetric, as there are no permutations to consider. A rank-1 tensor, equivalent to a vector, is likewise always symmetric because it has only a single index, leaving no pairs available for swapping while preserving the tensor's value under transformations. In Riemannian geometry, the metric tensor g_{ij} serves as a fundamental covariant example of a rank-2 symmetric tensor, satisfying g_{ij} = g_{ji} to define distances and angles consistently on a manifold. This symmetry ensures the metric functions as a non-degenerate inner product on tangent spaces. Contravariantly, the moment of inertia tensor I^{ij} in classical mechanics exemplifies symmetry, with I^{ij} = I^{ji}, arising from its construction as I^{ij} = \sum_\alpha m_\alpha (\delta^{ij} r_\alpha^2 - x_\alpha^i x_\alpha^j), where the coordinate products enforce equality under index exchange due to rotational invariance.^[7] For higher ranks, a simple example of a fully symmetric rank-3 tensor is the outer product v \otimes v \otimes v for a vector v, which remains unchanged under any permutation of its three indices.^[3]

Properties and Operations

Basic Properties

A symmetric tensor of order k on a vector space V is characterized by its invariance under the action of the symmetric group S_k, which permutes the tensor's indices or arguments. Specifically, the space of symmetric k-tensors, denoted S_k(V), is the image of the symmetrizer operator P = \frac{1}{k!} \sum_{\sigma \in S_k} \sigma, where \sigma acts by permuting the factors in the tensor product V^{\otimes k}. This operator is a projection onto S_k(V), satisfying P^2 = P, and ensures that any element T \in S_k(V) remains unchanged under any permutation: \sigma(T) = T for all \sigma \in S_k. This permutation invariance defines the algebraic structure of symmetric tensors and distinguishes them from general multilinear maps.^[4] For symmetric bilinear forms, which correspond to order-2 symmetric tensors in S_2(V^*), the symmetry implies that the associated form \langle u, v \rangle = \langle v, u \rangle for all u, v \in V. Over the real numbers, if this form is additionally positive definite—meaning \langle x, x \rangle > 0 for all nonzero x \in V—it defines an inner product on V, endowing the space with a Euclidean structure. This preservation of symmetry under argument swap is fundamental to the geometry induced by such tensors.^[8]^[8] The trace operation on a symmetric (0,2)-tensor T, defined as \operatorname{tr}(T) = T^i_i in a chosen basis (summing over repeated indices), is well-defined and independent of the basis due to the tensor's symmetry. This contraction yields a scalar invariant under coordinate changes, reflecting the tensor's intrinsic properties. Furthermore, orthogonal transformations preserve the symmetry of a tensor: if T is symmetric in one orthonormal basis, its components remain symmetric after any orthogonal change of basis, ensuring consistency across rotated frames.^[4]^[9] In representation theory, the space S_k(V) forms an irreducible representation of the general linear group \operatorname{GL}(V), meaning it contains no proper nontrivial invariant subspaces under the natural action of \operatorname{GL}(V). This irreducibility underscores the fundamental role of symmetric tensors in decomposing tensor spaces and highlights their algebraic independence from other symmetry types. The dimension of S_k(V) for \dim V = n is \binom{n+k-1}{k}, quantifying the degrees of freedom in such representations.^[4]

Symmetric Part of a Tensor

The symmetric part of a tensor is obtained by applying the symmetrization operator, which projects a general tensor onto the subspace of symmetric tensors by averaging over all permutations of its indices. For a rank-k tensor T \in V^{\otimes k} over a vector space V of dimension n, the symmetrization operator S: V^{\otimes k} \to V^{\otimes k} is defined as

S(T) = \frac{1}{k!} \sum_{\sigma \in S_k} \sigma(T),

where S_k is the symmetric group on k elements, and \sigma(T) denotes the action of the permutation \sigma by rearranging the tensor factors according to \sigma.^[4] This operator possesses key properties that make it a projection onto the symmetric subspace. Specifically, it is idempotent, satisfying S^2 = S, meaning applying S twice yields the same result as applying it once, and it commutes with the action of permutations, ensuring consistency under index relabeling.^[4] The image of S, denoted S_k(V), consists of all fully symmetric rank-k tensors, and its dimension is given by the binomial coefficient \binom{n+k-1}{k}, which counts the number of independent components.^[10] For mixed tensors, which have both contravariant and covariant indices, symmetrization can be applied partially over specific index sets rather than all indices. For instance, one may symmetrize only over a subset of contravariant indices by averaging over permutations within that group, while leaving other indices unchanged; the formula generalizes to S(T) = \frac{1}{m!} \sum_{\sigma \in S_m} \sigma(T) for an m-index subset, preserving the tensor's overall type.^[11] A concrete example illustrates this for a (0,2)-tensor A_{ij} in n dimensions: its symmetric part is

(A_{ij} + A_{ji})/2,

which averages over the two permutations of the covariant indices and reduces the independent components from n^2 to \binom{n+1}{2} = n(n+1)/2.^[10] Such operations preserve scalar invariants like the trace, as permutations do not alter the contraction along diagonal elements.^[4]

Symmetric Product

The symmetric product of two vectors v and w in a vector space V over a field F is defined as v \odot w = \frac{1}{2}(v \otimes w + w \otimes v), which lies in the space of symmetric bilinear forms or, equivalently, the second symmetric power S_2(V).^[4] This construction ensures that the result is invariant under interchange of v and w, yielding a symmetric rank-2 tensor. For higher ranks, the symmetric product generalizes through iterated applications: for vectors v_1, \dots, v_k \in V, the k-fold symmetric product is given by the symmetrizer \frac{1}{k!} \sum_{\sigma \in [S_k](/page/Symmetric_group)} v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(k)}, where S_k is the symmetric group on k elements, projecting into the k-th symmetric power S_k(V).^[4] The collection of all symmetric powers \bigoplus_{k=0}^\infty S_k(V) forms the symmetric algebra S(V), which is a graded commutative algebra under the extension of the symmetric product, with multiplication induced by the tensor product followed by symmetrization.^[12] In coordinates with respect to a basis of V, the basis elements of S(V) are monomials x_1^{a_1} \cdots x_n^{a_n} where \sum a_i = k for the degree-k component, reflecting the commutative structure where the product satisfies v \odot w = w \odot v.^[13] Symmetric tensors in S_k(V) are in natural correspondence with homogeneous polynomials of degree k on the dual space V^*, via the map that sends a symmetric tensor T to the polynomial p_T(u^*) = T(u^*, \dots, u^*) for u^* \in V^*, or more generally, p_T(u_1^*, \dots, u_k^*) = T(u_1^*, \dots, u_k^*) as a symmetric multilinear form.^[4] For example, a quadratic symmetric tensor corresponds to a quadratic form q(v) = T(v, v), which defines a bilinear form on V. This duality underpins the universal property of S_k(V): any symmetric k-linear map from V^k to another space extends uniquely to a linear map from S_k(V).^[4] The Veronese embedding provides a geometric realization of symmetric tensors, mapping the projective space \mathbb{P}(V) to \mathbb{P}(S_k(V)) via \nu_k() = [v^{\odot k}], where v^{\odot k} = v \odot \cdots \odot v (k times), embedding into a projective variety whose points correspond to rank-1 symmetric tensors. This map, of degree k, highlights the role of symmetric products in algebraic geometry, parametrizing hypersurfaces defined by homogeneous polynomials associated to higher-rank symmetric tensors.

Decompositions and Representations

Polar Decomposition

The polar decomposition theorem states that any invertible second-order tensor F, representing a linear map between vector spaces, can be uniquely factored as F = R U, where R is an orthogonal tensor satisfying R^T R = I and U is a positive definite symmetric tensor.^[14] This decomposition separates the rotational and stretching components of the transformation, with U capturing the pure deformation. The right stretch tensor U is explicitly given by U = \sqrt{F^T F}, which is always symmetric and positive definite for invertible F.^[14] A left polar form exists as F = V R, where V = \sqrt{F F^T} is the left stretch tensor, also positive definite and symmetric, while maintaining the same orthogonal R.^[14] Both forms are unique for invertible F, ensuring a canonical separation into orthogonal and symmetric positive definite factors. This applies specifically to (1,1)-tensors, or matrices in finite dimensions.^[14] For higher-rank tensors, the polar decomposition generalizes through higher-order forms that extend the matrix case via Kronecker-structured covariance matrices, decomposing a tensor X \in \mathbb{R}^{p_1 \times \cdots \times p_K \times n} into factors involving positive definite symmetric components across modes and an orthogonal core.^[15] In this extension, the symmetric stretch is captured by lower triangular matrices L_k with positive diagonals, forming positive definite P_k = L_k L_k^T, while the orthogonal part satisfies mode-wise orthonormality conditions. Uniqueness holds under conditions such as n \geq p, though practical computations often succeed more broadly.^[15] Computation for the second-order case proceeds via singular value decomposition (SVD) of F = Q \Sigma P^T, yielding U = P \Sigma P^T (with singular values as eigenvalues of U) and R = Q P^T.^[14] For higher ranks, an iterative block coordinate descent algorithm updates each mode's factor by applying matrix polar decomposition to unfolded slices, orthogonalizing the core simultaneously for stability.^[15] The eigenvalues of the symmetric stretch can be obtained via spectral decomposition of the relevant Gram tensors.^[14]

Spectral Decomposition

The spectral theorem for real symmetric tensors provides a diagonalization in an orthonormal basis, analogous to the matrix case. For a real symmetric (1,1)-tensor, which corresponds to a symmetric matrix T \in \mathbb{R}^{n \times n}, the theorem states that T can be decomposed as T = Q D Q^T, where Q is an orthogonal matrix whose columns are the eigenvectors of T, and D is a diagonal matrix containing the real eigenvalues \lambda_1, \dots, \lambda_n of T.^[16] This decomposition arises because symmetric matrices have only real eigenvalues, and their eigenvectors can be chosen to form an orthonormal basis, ensuring diagonalizability with no nontrivial Jordan blocks due to the symmetry.^[16] For higher-order symmetric tensors, the spectral decomposition generalizes to a multi-linear form using symmetric eigentensors. A symmetric tensor T \in S_d(\mathbb{R}^n) of order d \geq 3 admits a decomposition T = \sum_{i=1}^r \lambda_i u_i^{\otimes d}, where the \lambda_i are real eigenvalues associated with unit vectors u_i that are orthogonal (i.e., u_i^T u_j = \delta_{ij}) when the tensor is orthogonally decomposable, and r \leq n is the tensor rank.^[17] This extends the rank-2 case, with eigenvalues defined via the equation T u^{d-1} = \lambda u^{d-1} for H-eigenvalues (real eigenvectors) or T u^{d-1} = \lambda u for E-eigenvalues (normalized), both of which are real and invariant under orthogonal transformations.^[18] Multiplicity is handled by the number of distinct eigentensors, and the symmetry ensures orthogonality among them without complex Jordan-like structures.^[19] Computationally, for large symmetric tensors, the power iteration method iteratively approximates the dominant eigenvalue and eigenvector by updating u^{(k+1)} = T (u^{(k)})^{d-1} / \|T (u^{(k)})^{d-1}\|, converging linearly under suitable conditions like distinct leading eigenvalues.^[20] For rank-2 cases, the Lanczos algorithm efficiently computes extremal eigenvalues by tridiagonalizing the matrix through orthogonal transformations, offering faster convergence for sparse structures.^[21]

Young Tableau Representation

In representation theory, the structure of symmetric tensors is captured through Young diagrams, which classify the irreducible representations of both the symmetric group S_k and the general linear group \mathrm{GL}(V) for a vector space V of dimension n. For fully symmetric tensors of rank k, the corresponding Young diagram is the single-row partition $$, consisting of k boxes in one row; this labels the irreducible \mathrm{GL}(V)-representation S^{} V, which is precisely the k-th symmetric power \Sym^k V.^[22] The symmetric subspace within the tensor power V^{\otimes k} is obtained via the Young symmetrizer projector for the partition $$, defined as

P_{} = \frac{1}{k!} \sum_{\sigma \in S_k} \sigma,

where \sigma acts by permuting the tensor factors. This operator is the average over all elements of S_k and projects onto the invariant subspace under the symmetric group action, yielding the fully symmetric tensors; unlike antisymmetrizers, it involves no sign characters since $$ corresponds to the trivial representation of S_k.^[22] Schur-Weyl duality provides the full decomposition of the tensor power under the commuting actions of \mathrm{GL}(V) and S_k:

V^{\otimes k} \cong \bigoplus_{\lambda \vdash k} S^\lambda V \otimes U^\lambda,

where the direct sum runs over all partitions \lambda of k, S^\lambda V is the irreducible \mathrm{GL}(V)-module (Schur module) labeled by \lambda, and U^\lambda is the corresponding irreducible S_k-module (Specht module). The symmetric tensors form the summand for \lambda = , where S^{} V is the symmetric subspace tensored with the one-dimensional trivial S_k-representation.^[23] The dimension of this symmetric representation is

\dim(S^{} V) = \binom{n + k - 1}{k},

which counts the number of monomials of degree k in n variables and arises as a special case of the Weyl dimension formula for Schur modules. For general partitions \lambda \vdash k, the dimension of S^\lambda V is given by the hook-content formula:

\dim(S^\lambda V) = \prod_{(i,j) \in \lambda} \frac{n + c(i,j)}{h(i,j)},

where the product is over boxes in the Young diagram of \lambda, c(i,j) = j - i is the content of box (i,j), and h(i,j) is its hook length (the number of boxes to the right and below, plus one for the box itself). In the symmetric case \lambda = , each of the k boxes in the single row has hook length k - j + 1 for the j-th box, reducing the formula to the binomial coefficient \binom{n + k - 1}{k}; the underlying hook-length formula f^\lambda = k! / \prod h(i,j) originally computes the dimension of the S_k-irreducible U^\lambda.

Applications

In Physics

In physics, symmetric tensors play a crucial role in describing conservation laws and physical observables across various domains, from classical mechanics to quantum field theory and general relativity. Their symmetry often arises from fundamental symmetries of the underlying physical laws, ensuring consistency with principles like angular momentum conservation. The stress-energy tensor T^{\mu\nu}, which encodes the distribution of energy, momentum, and stress in spacetime, is symmetric (T^{\mu\nu} = T^{\nu\mu}) due to the conservation of angular momentum derived from Noether's theorem applied to Lorentz invariance. This symmetry implies that the total angular momentum, including contributions from orbital and spin parts, is conserved in isolated systems, a cornerstone of relativistic field theories. In the absence of spin, the canonical stress-energy tensor derived from the Lagrangian is automatically symmetric; for fields with intrinsic spin, a symmetrized version (e.g., Belinfante-Rosenfeld tensor) restores this property to match observations.^[24] In classical mechanics, the inertia tensor I_{ij} for a rigid body is a symmetric second-rank tensor that quantifies rotational inertia about the center of mass, given by I_{ij} = \int (r^2 \delta_{ij} - x_i x_j) \, dm, where r is the distance from the axis, \delta_{ij} is the Kronecker delta, and the integral is over the mass distribution. This symmetry follows from the scalar nature of the moment of inertia and ensures that the angular momentum \mathbf{L} = \mathbf{I} \cdot \boldsymbol{\omega} and torque \boldsymbol{\tau} = \mathbf{I} \cdot \dot{\boldsymbol{\omega}} equations are well-defined for rigid body dynamics, simplifying the Euler equations for rotation.^[25] In electromagnetism, the electromagnetic field strength tensor F_{\mu\nu} is antisymmetric (F_{\mu\nu} = -F_{\nu\mu}), capturing the duality between electric and magnetic fields, whereas the associated electromagnetic stress-energy tensor remains symmetric to align with the general requirements of energy-momentum conservation. This contrast highlights how symmetry properties distinguish field strengths from their energy-momentum contributions, with the symmetric stress-energy tensor ensuring compatibility with Noether-derived conservation laws.^[26] In quantum mechanics, the density matrix \rho represents the statistical state of a quantum system as a symmetric (Hermitian) operator that is positive semidefinite with trace unity (\operatorname{Tr}(\rho) = 1), allowing computation of expectation values \langle A \rangle = \operatorname{Tr}(\rho A) for observables A. This form generalizes pure states (\rho = |\psi\rangle\langle\psi|) to mixed states, preserving probabilistic interpretations and enabling descriptions of open quantum systems and entanglement.^[27] In general relativity, the Einstein tensor G_{\mu\nu}, defined as G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}, is symmetric due to the contracted Bianchi identities, which enforce the covariant conservation \nabla^\mu G_{\mu\nu} = 0. This symmetry ensures that the Einstein field equations G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} couple consistently to the symmetric stress-energy tensor, upholding energy-momentum conservation in curved spacetime. The metric tensor g_{\mu\nu}, also symmetric, provides the background geometry for these relations.^[28]^[29]

In Differential Geometry

In differential geometry, symmetric tensors play a fundamental role on Riemannian manifolds, where they often arise as sections of tensor bundles that encode geometric structures invariant under certain symmetries. A key example is the metric tensor, which defines the geometry of the manifold. On a smooth manifold M, a Riemannian metric g is a smooth section of the bundle of symmetric bilinear forms on the tangent bundle, specifically a (0,2)-tensor that is symmetric (g(X,Y) = g(Y,X) for vector fields X, Y) and positive definite at each point, thereby inducing an inner product on each tangent space T_p M. This symmetry ensures that the metric preserves the inner product structure under parallel transport along geodesics, facilitating the measurement of lengths, angles, and volumes on the manifold. The Levi-Civita connection, uniquely determined by the metric tensor, introduces symmetric tensors through its connection coefficients, known as Christoffel symbols. These symbols \Gamma^k_{ij} are derived from the metric via the formula

\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \right),

where g^{kl} is the inverse metric, and \partial denotes partial differentiation in local coordinates. The symmetry of the metric g_{ij} = g_{ji} implies that the Christoffel symbols are symmetric in their lower indices, \Gamma^k_{ij} = \Gamma^k_{ji}, which corresponds to the torsion-free property of the Levi-Civita connection. This symmetry is crucial for defining covariant derivatives that are compatible with the metric, enabling the extension of differentiation to tensor fields while preserving the inner product. Another important symmetric tensor is the Hessian, which captures second-order geometric information. For a smooth function f: M \to \mathbb{R} on a Riemannian manifold, the Hessian \nabla^2 f is defined as the second covariant derivative, \nabla^2 f (X, Y) = X(Y f) - (\nabla_X Y) f, and it forms a symmetric (0,2)-tensor field acting as a bilinear form on tangent vectors. The symmetry \nabla^2 f (X, Y) = \nabla^2 f (Y, X) follows from the torsion-freeness of the connection and the metric compatibility, making the Hessian a natural tool for studying critical points, geodesics, and curvature variations. In local coordinates, its components are H_{ij} = \partial_i \partial_j f - \Gamma^k_{ij} \partial_k f, highlighting its role in the geometry of submanifolds and optimization on manifolds. The Weyl tensor, a conformally invariant component of the curvature, also exhibits symmetries involving symmetric parts in its index structure. Defined as the trace-free part of the Riemann curvature tensor, W^a_{bcd} = R^a_{bcd} - \frac{1}{n-2} (R^a_c g_{bd} - R^a_d g_{bc}) + \frac{R}{(n-1)(n-2)} (g^a_c g_{bd} - g^a_d g_{bc}) in n-dimensions, the Weyl tensor inherits the Riemann tensor's symmetries: antisymmetry in the last two indices (W^a_{bcd} = -W^a_{bdc}), antisymmetry in the first pair when raised (W_{abcd} = -W_{bacd}), and symmetry under pair exchange (W_{abcd} = W_{cdab}). These symmetries, including the symmetric interchange of index pairs, ensure that the Weyl tensor measures the intrinsic conformal distortion of the manifold, independent of local scaling.^[30] Conformal transformations further underscore the role of symmetric tensors by preserving their symmetry properties up to scaling. A conformal change of metric is given by \tilde{[g](/page/G)} = e^{2\sigma} [g](/page/G), where \sigma is a smooth function on M, transforming the original symmetric metric [g](/page/G) to another symmetric metric \tilde{[g](/page/G)} since the exponential factor is a scalar multiple that commutes with the symmetry. This rescaling leaves angles invariant while altering lengths, and the Weyl tensor remains unchanged under such transformations, highlighting how symmetric tensors like the metric maintain their structural integrity in conformal geometry.

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