15.17.2 Tensor Symmetric Component Transformation

Tensor Symmetric Component Transformation reorganizes symmetric tensor parts via algebraic methods, maintaining symmetry for coordinate-independent analysis.

Tensor Symmetric Component Transformation is the derivation, from first principles of multilinearity and basis expansion, of the precise formula by which the individual numerical components of a symmetric tensor change when the underlying vector space is re-coordinatized, tracing the formula back to the definition of a tensor as a multilinear object rather than simply stating it as a rule to be applied.

Deriving the Formula from Multilinearity

Tensors as Multilinear Functionals in Coordinates

A tensor T of order d on a vector space V can be regarded as a multilinear functional taking d vector arguments and returning a scalar. Once a basis e_1 through e_n of V is fixed, the components of T are defined as its values on all possible tuples of basis vectors:

T_{i_{1} \dots i_{d}} = T (e_{i_{1}}, \dots, e_{i_{d}})

The Component Transformation formula is obtained by expressing a new basis vector e'_i as a linear combination of the old basis vectors, using the entries of the change-of-basis matrix P, and substituting this expansion into every one of the d argument slots of T.

Substitution Slot by Slot

Writing the new basis vector as e'_i equal to the sum over j of P_ji times e_j, and substituting this expression into each of the d arguments of T in turn, multilinearity allows each substitution to be pulled outside the functional evaluation as a separate sum with its own coefficient, one factor of P per index slot, leaving the original components of T evaluated on old basis vectors inside. Collecting all d substitutions together produces exactly the transformation formula stated in the general Transformation Behavior,

{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}}

confirming that this formula is not an ad hoc rule but a direct consequence of what it means for T to be multilinear.

Role of Variance Convention

Covariant Components

The derivation above applies directly when T is treated as a functional of vectors, in which case its components are called covariant, and they transform using the matrix P itself, as shown, which describes the new basis vectors in terms of the old ones.

Contravariant Components

When the underlying object is instead built from vectors directly, such as a tensor representing a bivector or a linear map's target space, its components are called contravariant and transform instead using the inverse of P, since the coordinates of a fixed vector change inversely to how the basis vectors themselves change, in order to keep the vector itself unchanged. The Component Transformation formula for the fully symmetric, fully covariant tensors central to Tensor Symmetric Decomposition Structure always uses P in the form shown above, but recognizing the alternative contravariant convention clarifies why some tensor formulas elsewhere use P-inverse instead.

Explicit Low-Order Illustrations

Order One: Ordinary Vector Components

For d equal to one, the Component Transformation formula reduces to the single-sum expression T'_i equal to the sum over j of P_ji times T_j, which is exactly the ordinary rule for how the coordinates of a covector change under a change of basis, confirming the formula's consistency with the most elementary, already-familiar case.

Order Two: Recovering the Congruence Rule

For d equal to two, the double-sum Component Transformation formula is precisely the entrywise expansion of the matrix congruence rule T' equal to P-transpose T P, so that the abstract multilinear derivation given here reproduces, as a special case, the matrix formula used throughout the Matrix Case and its Diagonalization Context.

Preservation of the Symmetric Structure Through the Derivation

Symmetry Survives Because the Derivation Treats All Slots Alike

Because the substitution procedure used to derive the Component Transformation formula treats every one of the d argument slots of T identically, applying the same substitution rule regardless of which slot is being processed, any permutation symmetry present among the original components carries through unchanged to the new components; this is the constructive, derivation-based counterpart of the abstract Transformation Preservation theorem, showing concretely, rather than merely asserting, why symmetry is never disturbed by a change of coordinates.

Consistency Across the Theory

Because the Component Transformation formula is derived directly from multilinearity, it applies uniformly to every symmetric tensor regardless of order, underlying the Basis Change Response computations carried out explicitly for particular matrices, and underlying the invariance of symmetric rank, apolarity, and every other basis-independent quantity studied throughout the theory of symmetric tensors.