15.19 Tensor Symmetric Tensor Notation

Tensor Symmetric Tensor Notation is the collection of notational conventions used to write, manipulate, and communicate symmetric tensors and their operations, spanning explicit indexed component notation, symmetrization brackets, abstract coordinate-free notation, and the polynomial notation inherited from the correspondence with homogeneous forms.

Indexed Component Notation

Basic Index Placement

The most explicit notational convention writes a symmetric tensor T of order d with d lower indices, T subscript i_1 through i_d, each index ranging independently over the dimension of the underlying vector space, following the general convention for covariant tensor components. Because symmetric tensors arise most naturally as fully covariant objects, in the sense discussed under the Tensor Role of the symmetric matrix, this all-subscript placement is standard throughout the notation used for symmetric tensor components, in contrast with the mixed upper-and-lower index notation required for general tensors that mix vector and covector slots.

Explicit Symmetry Markers

To signal that a given expression is understood to be symmetric without repeating the full permutation-invariance condition, parentheses are placed around the indices being symmetrized:

which is the notational device for the symmetrization operator applied to an arbitrary, not necessarily symmetric, tensor's indices, averaging over all permutations with the normalizing factor of one over d factorial.

Abstract, Coordinate-Free Notation

Symmetric Power Notation

The space of symmetric tensors of order d on a vector space V is denoted using the symmetric power symbol, S superscript d, applied to V, written S^d V, distinguishing it from the full, unrestricted tensor power V superscript tensor d, and this notation directly encodes the containment of the symmetric tensors as a distinguished subspace, consistent with the Subspace Invariance established for this subspace under change of basis.

Pure Power Form Notation

A pure power form, the d-th tensor power of a single vector v, is written v superscript tensor d, using the same tensor-power symbol applied directly to the vector rather than to the space, and this notation is the standard shorthand used throughout the description of symmetric decompositions, appearing repeatedly in the defining sum of the Tensor Symmetric Decomposition Rank Relation and in the Term Set concept.

Notation for Decomposition and Rank

Rank Subscript Notation

To distinguish the symmetric rank of a tensor from its ordinary tensor rank, a subscript S is appended to the rank symbol, giving rank subscript S of T, as used throughout the Rank Relation, while the unadorned rank of T denotes the ordinary, unrestricted tensor rank; this subscripted convention is essential precisely because the two quantities can differ, and the notation must therefore never conflate them.

Sum-of-Powers Notation for Decompositions

A symmetric decomposition is conventionally written as an explicit finite sum of pure power forms with an index running from one to the claimed rank, exactly the notation used in the defining formula of the Term Set concept, and this sum notation is understood to carry the implicit claim that the number of terms displayed is minimal whenever it is asserted to compute the symmetric rank rather than merely to exhibit some valid decomposition.

Notation Inherited from the Polynomial Correspondence

Homogeneous Polynomial Notation

Because symmetric tensors of order d correspond to homogeneous polynomials of degree d, symmetric tensors are frequently written directly in polynomial notation, using ordinary variables x_1 through x_n in place of tensor indices, as illustrated throughout the Tensor Quadratic Form Component Expression; this dual notation, switching freely between indexed tensor components and polynomial coefficients, is standard practice and relies on the basis-fixed correspondence between the two descriptions.

Waring Rank Notation

When working in the polynomial notation, the symmetric rank is often referred to instead as the Waring rank of the form, and decompositions are written as sums of powers of linear forms rather than as sums of tensor powers of vectors, a purely notational, not substantive, difference from the tensor-theoretic sum-of-powers notation described above.

Matrix Notation as a Special Case

Order-Two Simplification

For order two, the general indexed notation simplifies to ordinary matrix notation, with T written as a boldface or capital letter denoting the full array, and operations such as the congruence transformation written using matrix multiplication and the transpose symbol rather than explicit indices and summation signs, exactly as used throughout the Matrix Case and its associated Diagonalization Context; this simplification is available specifically because order-two tensors admit the additional structure of matrix multiplication, absent for tensors of order three or higher.

Consistency of Notation Across the Theory

Notation as a Tool for Tracking Structure, Not an End in Itself

Every notational device described here, from index parentheses to symmetric power symbols to polynomial variables, exists to make a specific structural fact about symmetric tensors easier to state and manipulate correctly; the choice among them in any given context is guided entirely by which notation most clearly exposes the relevant structure, whether that is the permutation-invariance emphasized by indexed notation, the subspace structure emphasized by symmetric power notation, or the computational tractability emphasized by matrix notation in the order-two case.