15.17.1 Tensor Symmetric Transformation Preservation

Tensor Symmetric Transformation Preservation ensures symmetry is maintained under transformations, crucial in algebraic structures and tensor theory applications.

Tensor Symmetric Transformation Preservation is the theorem, together with its proof, stating that the property of being a symmetric tensor is invariant under every admissible change of basis, so that the subspace of symmetric tensors within the full tensor product space is left setwise unchanged by the action of any invertible linear transformation of the underlying vector space.

Statement of the Preservation Theorem

Precise Formulation

Let T be a tensor of order d on a vector space V, and let P be any invertible linear transformation of V. If T satisfies the Component Constraint, meaning its components are unchanged under every permutation of the d index positions, then the transformed tensor T', obtained by applying the general Transformation Behavior formula with respect to P, also satisfies the Component Constraint. In symbols, if

T_{i_{1} \dots i_{d}} = T_{i_{σ (1)} \dots i_{σ (d)}}

for every permutation sigma and every choice of indices, then the same equality holds with T replaced by T' throughout.

Reformulation as Subspace Invariance

Equivalently, the theorem states that the space of symmetric tensors of order d, viewed as a linear subspace of the full tensor product space V tensor d, is invariant under the natural action of the general linear group of V on that tensor product space: applying any invertible linear map to a symmetric tensor, via the standard action on tensors, always produces another symmetric tensor, never a non-symmetric one.

Proof of Preservation

Direct Verification via Index Substitution

The transformation formula expresses each component of T' as a sum, over all choices of d auxiliary indices j_1 through j_d, of a product of d entries of P (one for each index slot) times the corresponding component of T:

{T^{'}}_{i_{1} \dots i_{d}} = \sum_{j_{1} \dots j_{d}}^{} P_{j_{1} i_{1}} \dots P_{j_{d} i_{d}} T_{j_{1} \dots j_{d}}

To check invariance under a permutation sigma of the output indices i_1 through i_d, permute both the output indices on the left and, correspondingly, relabel the summation indices j_1 through j_d on the right by the same permutation. Because the sum ranges over all values of every j_k independently, this relabeling changes nothing about the value of the sum; what it does accomplish is to move the entries of P into a new order matching the permuted output indices, and to move the summed indices of T into the same permuted order. Since T is already symmetric, its component with the permuted index order equals its component with the original index order, and substituting this equality shows the permuted expression for T' equals the original expression for T', which is exactly the required invariance.

Why the Argument Requires No Special Property of P

Crucially, the proof uses no property of P beyond its entries being well-defined scalars: invertibility of P is not needed for this particular preservation statement (it is needed only to ensure the transformation is reversible), and no assumption of orthogonality, or any other special structure, is required. This is why Transformation Preservation holds unconditionally for every change of basis, in sharp contrast to properties like diagonalizability or eigenvalue behavior, which do depend on the specific type of P chosen, as seen in the varying Basis Change Response to orthogonal, scaling, and shear transformations.

Consequences of Preservation

Well-Definedness of the Symmetric Tensor Space

Transformation Preservation is precisely what justifies speaking of "the space of symmetric tensors" as an intrinsic, coordinate-free object, since without this guarantee, whether a given abstract tensor counts as symmetric might depend on which basis was used to check the Component Constraint, and the theorem rules this out: symmetry, once established in any one basis, holds in every basis.

Foundation for Rank and Decomposition Theory Being Basis-Independent

Every subsequent invariant discussed throughout Tensor Symmetric Decomposition Structure, including symmetric rank, border rank, and the classification via secant varieties of the Veronese variety, is defined using the intrinsic space of symmetric tensors as its domain; Transformation Preservation is the theorem that makes this domain well-defined, and it is silently relied upon whenever a computation is carried out in a convenient basis and the resulting rank or decomposition is asserted to hold for the original tensor in any basis.

Compatibility with the Symmetrization Projector

Transformation Preservation also confirms the consistency of the symmetrization operator, which projects a general tensor onto its symmetric part: because the image of this projector already lies in the invariant subspace of symmetric tensors, applying a change of basis before or after symmetrization produces the same result, a compatibility used implicitly whenever the symmetric part of a tensor product, as in the construction of higher-order symmetric tensors from the Symmetric Tensor Role of a quadratic form, is computed in a chosen coordinate system.