15.2 Tensor Symmetric Tensor Areas

Tensor Symmetric Tensor Areas is the survey of the principal mathematical and applied domains in which symmetric tensors arise as the natural carrier of the relevant data, spanning quadratic and bilinear forms, differential geometry, classical mechanics, and probability and statistics, each of which relies on the full permutation invariance of a symmetric tensor for a different underlying reason.

Quadratic and Bilinear Forms

Symmetric Bilinear Forms as Degree-Two Symmetric Tensors

A symmetric bilinear form $B:V\times V\to K$, satisfying $B(v,w)=B(w,v)$, corresponds precisely to an element of ${\text{Sym}}^2\left(V^{\ast }\right)$, and the associated quadratic form $Q\left(v\right)=B(v,v)$ recovers $B$ completely whenever the characteristic of the field is not two, making symmetric tensors of degree two the natural home for the theory of quadratic forms, diagonalization by orthogonal or congruence transformations, and signature classification.

Differential Geometry

The Metric Tensor

A Riemannian or pseudo-Riemannian metric is, at each point of a manifold, a symmetric bilinear form on the tangent space, varying smoothly from point to point; it is a symmetric tensor field precisely because distances and angles must not depend on an arbitrary ordering of the two tangent vectors being compared.

The Ricci and Related Curvature Tensors

Several of the fundamental curvature tensors built from a metric, most notably the Ricci curvature tensor, are symmetric in their two indices, a symmetry that follows from deeper identities of the full Riemann curvature tensor and that has direct consequences for the structure of the Einstein field equations in general relativity, where the stress-energy tensor on the other side of the equation is likewise required to be symmetric.

Classical Mechanics

The Inertia Tensor

The moment of inertia tensor of a rigid body, relating angular velocity to angular momentum, is a symmetric degree-two tensor, with its symmetry reflecting the fact that the kinetic energy of rotation, a quadratic form built from this tensor, must be well defined regardless of how the two velocity arguments are labeled.

Stress and Strain Tensors

The stress tensor and the strain tensor of continuum mechanics, describing internal forces and deformations within a material, are symmetric under the standard assumption of local equilibrium with no internal body torques, so that symmetric tensors provide the natural mathematical description of both the cause and the effect of mechanical deformation in a continuous medium.

Probability and Statistics

Covariance Matrices as Symmetric Tensors

The covariance matrix of a random vector, recording the covariance between every pair of its components, is symmetric because covariance itself is a symmetric bilinear quantity, $\text{Cov}(X,Y)=\text{Cov}(Y,X)$, placing covariance matrices squarely within the theory of degree-two symmetric tensors and connecting their diagonalization directly to principal component analysis.

Higher Cumulants and Moments

Moments and cumulants of a multivariate distribution beyond the second order are indexed by more than two coordinates and are symmetric in all of them, since the joint moment $E[X_{i_1}\cdots X_{i_k}]$ does not depend on the order in which the random variables are multiplied, making the full higher-order moment and cumulant tensors natural elements of ${\text{Sym}}^k$ for the corresponding degree.

Polynomial and Invariant Theory

Homogeneous Polynomials as Symmetric Tensor Data

As established at the level of the symmetric algebra, every homogeneous polynomial of degree $k$ corresponds to a symmetric tensor of the same degree, making symmetric tensors the natural setting for classical invariant theory, which studies how the coefficients of such polynomials transform under a change of variables, and for questions about the rank and decomposition of polynomials as sums of powers of linear forms.