15.17 Tensor Symmetric Transformation Behavior

Tensor Symmetric Transformation Behavior is the description of how the components of a symmetric tensor of any order change under a change of basis of the underlying vector space, and of the guarantee that this change of coordinates always preserves the defining symmetry, so that the symmetric tensors form a well-defined, basis-independent geometric object rather than a merely coordinate-dependent bookkeeping device.

The General Transformation Rule

Transformation of an Order-d Tensor

A tensor T of order d on an n-dimensional vector space is represented, once a basis is chosen, by components indexed by d indices, each ranging over the n basis directions. Under a change of basis given by an invertible matrix P, whose columns express the new basis vectors in terms of the old, the components of T transform by applying P (or its inverse, depending on variance convention) once along each of the d index slots simultaneously:

This multilinear transformation rule is the direct generalization, to d index slots, of the congruence transformation used for order-two symmetric tensors under the Tensor Role of a symmetric matrix, and it reduces exactly to that congruence rule when d equals two.

Basis-Independence of the Tensor Itself

Although the numerical array of components changes under this rule, the tensor T, understood abstractly as a multilinear functional or as an element of the appropriate tensor product space, does not change; the transformation rule exists precisely to describe how the same underlying object is represented differently by different observers, and this is the standard meaning of tensoriality shared by symmetric tensors with all other tensors, symmetric or not.

Preservation of Symmetry Under the Transformation

Why Symmetry Survives a Change of Basis

The defining property of a symmetric tensor, invariance of its components under any permutation of the index positions, is preserved by the general transformation rule for every choice of P: applying the same permutation to the index positions on both sides of the transformation formula, and using that the components of T on the right-hand side are already permutation-invariant by hypothesis, shows that the transformed components on the left-hand side are permutation-invariant as well. Symmetry, unlike an accidental numerical coincidence among components, is therefore a property that survives any change of basis, confirming that "being a symmetric tensor" is a basis-independent, geometric condition rather than a property of one particular coordinate presentation.

Contrast with Non-Tensorial Symmetric Arrays

An array of numbers that merely happens to satisfy the Component Constraint in one basis, but is not the coordinate representation of a genuine multilinear object, will not, in general, continue to satisfy the constraint after an arbitrary change of coordinates applied incorrectly (for instance, by a rule appropriate to a different tensor valence). The Transformation Behavior is what distinguishes true symmetric tensors, whose symmetry is guaranteed to persist under the correct multilinear transformation law, from superficially symmetric-looking numerical tables that do not represent tensors at all.

Behavior Under Restricted Classes of Basis Change

Orthogonal Transformations

When the change of basis matrix P is restricted to be orthogonal, the general transformation rule specializes to a form in which the inverse of P coincides with its transpose, and this restricted setting is precisely the one in which the spectral and diagonalization results of the Matrix Case become available for order-two symmetric tensors; for higher-order symmetric tensors, orthogonal transformations still preserve symmetry, and they preserve additional structure such as the Frobenius-type norm of the tensor, but they do not, in general, produce a diagonal form analogous to the order-two spectral theorem.

Scaling and the Behavior of Pure Power Forms

Under a change of basis that is a pure scaling, the pure power form of a vector v transforms simply by the corresponding power of the scaling factor applied to each coordinate, consistent with the general rule and confirming, at the simplest possible instance, that pure power forms transform correctly as symmetric tensors, a fact used implicitly throughout Tensor Symmetric Decomposition Structure whenever decompositions are compared before and after a change of coordinates.

Consequences for Invariant Theory

Invariance of Tensor Rank and Symmetric Rank

Because the Transformation Behavior guarantees that a change of basis acts on the space of decompositions of T by acting termwise on each vector of the term set (an invertible linear map applied to each v_i individually), the symmetric rank of a tensor, being defined as a minimum over decompositions, is unchanged by any change of basis; this invariance is what allows the Rank Relation and the classification results built on the Alexander-Hirschowitz theorem to be stated as intrinsic facts about a tensor, rather than as facts contingent on a particular coordinate presentation.

Invariance of Apolarity and Catalecticant Rank

The apolar ideal and the ranks of catalecticant matrices associated with T, used throughout Reconstruction and Term Set analysis, transform consistently under the general Transformation Behavior, since a change of basis on the underlying vector space induces a corresponding, invertible change of variables on the dual space of linear forms, under which vanishing of a differential operator on T is preserved; this consistency is what allows apolarity-based methods to be applied in any convenient coordinate system without affecting the resulting rank or decomposition conclusions.