15.17.3 Tensor Symmetric Basis Change Response

Exploring how symmetric basis transformations influence tensor responses within algebraic frameworks and their structural implications.

Tensor Symmetric Basis Change Response is the concrete, step-by-step account of what actually happens to the numerical components of a symmetric tensor when a specific change of basis is applied, worked out explicitly enough to be carried out by hand or by direct computation, in contrast to the general, index-notation transformation law from which it is derived.

Setting Up the Response Computation

Old and New Coordinates

Suppose a symmetric tensor T of order d is known in an old basis, with components T_{i1...id}, and a new basis is introduced, related to the old one by an invertible matrix P, whose columns give the new basis vectors expressed in the old coordinates. The Basis Change Response is the explicit computation of the new components T'_{i1...id}, obtained by substituting the relation between old and new coordinates into every occurrence of an index in the general Transformation Behavior formula.

Order-Two Worked Response

For a symmetric matrix T, the response to the basis change P is computed in two stages: first, right-multiply T by P, producing an intermediate array T P; second, left-multiply the result by P-transpose, producing the final response,

T^{'} = P^{transpose} (T P)

Carrying out this two-stage computation explicitly, entry by entry, is the Basis Change Response for the order-two case, and it can be checked directly, by expanding the matrix products, that the resulting array T' again satisfies the Component Constraint of symmetry for every choice of P.

Response to Specific Types of Basis Change

Response to an Orthogonal (Rotation-Type) Change

When P is orthogonal, so that P-transpose equals P-inverse, the Basis Change Response for an order-two symmetric tensor becomes a similarity transformation as well as a congruence transformation simultaneously, and the response preserves the eigenvalues of T exactly, only rotating the associated eigenvectors; this is the response case relevant to the Diagonalization Context, where an orthogonal P is sought specifically because it produces the simplest possible response, namely a purely diagonal result.

Response to a Scaling Change

When P is a diagonal matrix with entries c_1 through c_n (a coordinate-wise rescaling), the response of each component T_{i1...id} is to be multiplied by the product of the scaling factors corresponding to its index positions,

{T^{'}}_{i_{1} \dots i_{d}} = c_{i_{1}} \dots c_{i_{d}} T_{i_{1} \dots i_{d}}

which is the simplest possible nontrivial instance of the response, since no mixing between different index directions occurs, and it directly generalizes the response of a single pure power form under rescaling of its generating vector.

Response to a Shear (Non-Orthogonal) Change

When P is a non-orthogonal invertible matrix, such as a shear that adds a multiple of one coordinate direction to another, the Basis Change Response mixes previously independent components together: entries of T' generally become linear combinations of several entries of T, and the resulting array, while still symmetric, may look qualitatively different from T, for instance losing a diagonal or sparse structure that T originally possessed. This sensitivity to the specific choice of P is why the Diagonalization Context restricts attention to orthogonal (or, more generally, congruence) changes of basis chosen deliberately to produce a favorable response.

Response of Associated Quantities

Response of the Pure Power Form

If T is itself a single pure power form, v tensor-power d, its response to a basis change given by P is again a pure power form, generated by the vector obtained by applying P-inverse (or P, depending on whether v's coordinates are treated as contravariant or covariant) to v; this is the mechanism by which the Term Set of a symmetric decomposition responds to a change of basis, with each vector of the term set individually transformed by the same rule applied to v.

Response of the Symmetric Rank

Because the Basis Change Response acts on a decomposition by transforming each term individually and invertibly, the number of terms in a minimal decomposition cannot change: the symmetric rank of T' is identical to the symmetric rank of T, confirming, at the level of an explicit worked response, the basis-independence of rank asserted more abstractly under the general Transformation Behavior.

Practical Use of the Response Computation

Simplifying a Tensor by Choosing a Favorable Response

In practice, the Basis Change Response is computed deliberately, in reverse: rather than being given P and asked for T', one seeks a P whose response to a given T is as simple as possible, for instance diagonal, sparse, or normalized to have unit-length terms; this search is exactly the computational core of the Diagonalization Context for order two and of Reconstruction methods based on simultaneous diagonalization for higher orders.

Verifying Tensorial Claims by Direct Computation

Whenever a general claim about symmetric tensors, such as preservation of symmetry or invariance of rank, is in doubt, computing the explicit Basis Change Response for a small, concrete example provides a direct, checkable confirmation, serving as the elementary computational counterpart to the abstract, index-notation proofs given under the general Transformation Behavior.