15.19.5 Tensor Symmetric Basis Notation

Tensor Symmetric Basis Notation is the explicit notational scheme for writing down a basis of the space of symmetric tensors, built from symmetric products of the underlying vector space's basis vectors and labeled by the same multi-indices used in the Tensor Symmetric Index Notation, together with the normalization choices that determine whether this basis is orthonormal with respect to a natural inner product.

Constructing the Basis

Basis Vectors Labeled by Multi-Indices

Given a basis e_1 through e_n of V, a basis for the symmetric power S^d V is obtained by forming the symmetric product, in the sense of the Symmetric Product Notation, of d basis vectors chosen with repetition, one basis element for each multi-index alpha with entries summing to d:

where each factor e_j raised to the symmetric power alpha_j denotes the pure power form of e_j repeated alpha_j times. Because the multi-indices alpha range exactly over all non-negative integer tuples summing to d, and because distinct multi-indices correspond to genuinely independent symmetric tensor components, the collection e_alpha forms a basis of S^d V, matching the dimension count given by the binomial coefficient of n plus d minus one, choose d.

Expressing an Arbitrary Symmetric Tensor in This Basis

Any symmetric tensor T in S^d V can be written uniquely as a linear combination of the basis elements e_alpha, with coefficients given by the multi-index components T_alpha introduced under the Symmetric Index Notation:

with the sum ranging over all multi-indices alpha of total degree d, giving the Symmetric Basis Notation its role as the concrete expansion underlying the abstract multi-index component notation.

Normalization Choices

The Unnormalized Basis

Constructed directly as above, without any additional scaling, the basis elements e_alpha are not, in general, orthonormal with respect to the inner product on S^d V induced naturally from an inner product on V, since the symmetric product construction introduces multiplicities that depend on the multinomial coefficient associated with alpha, causing basis vectors corresponding to multi-indices with repeated entries to have different induced norms than those corresponding to multi-indices with all distinct entries.

The Normalized, Orthonormal Basis

An orthonormal basis is instead obtained by rescaling each e_alpha by the square root of the appropriate multinomial coefficient,

and this rescaled family is orthonormal precisely when the original basis e_1 through e_n of V was itself orthonormal, a fact that underlies the standard convention, in numerical and geometric applications, of working with the normalized basis whenever distances, angles, or norms of symmetric tensors are of direct interest.

Use of the Basis in Computation

Reading Off Coefficients via the Inner Product

When the orthonormal, normalized basis is used, the coefficient of a symmetric tensor T along a given basis direction is obtained directly as the inner product of T with the corresponding normalized basis vector, exactly mirroring the familiar procedure for reading off coordinates in an orthonormal basis of an ordinary vector space, and this is the mechanism by which catalecticant matrices and other coordinate-dependent constructions used throughout apolarity and Reconstruction are assembled explicitly from a symmetric tensor's abstract description.

Matching the Basis to the Polynomial Correspondence

Under the identification of S^d V with homogeneous polynomials, the unnormalized basis vector e_alpha corresponds to the monomial with exponents given by alpha, while the normalized basis vector corresponds to that same monomial scaled by the square root of its multinomial coefficient; the choice between these two conventions is the same choice, familiar from the study of orthogonal polynomials and spherical harmonics, between working with monomials directly or with a rescaled, norm-adapted family better suited to orthogonality-based computations.

Relationship to Other Notational Devices

Compatibility with Symmetric Product and Index Notation

The Symmetric Basis Notation is built entirely from the Symmetric Product Notation, applied to repeated basis vectors, and labeled using the same multi-indices introduced under the Symmetric Index Notation; it adds no new mathematical content beyond these two conventions but supplies the single, concrete spanning set that makes explicit numerical or symbolic computation with elements of S^d V possible, playing the same foundational role for symmetric tensor spaces that a chosen basis plays for any finite-dimensional vector space.