9.14.4 Tensor Basis Change Component Update

Understanding how tensor components change when the basis is altered in tensor algebra.

Tensor Basis Change Component Update is the specific step within a basis change operation in which the numerical values of a tensor's components are recalculated to reflect the new basis, replacing the old component array with a new one while leaving the tensor itself unaffected. It is the concrete computational action that carries out what the basis change matrix relation prescribes.

What the Update Accomplishes

Recalculating, Not Redefining

The component update takes the existing components, defined relative to the source system, and computes new numerical values, defined relative to the target system, using the transformation matrix and its inverse as required by the tensor's index pattern. The tensor being described does not change; only the numbers used to describe it do.

Applying to Every Component Slot

The update must be carried out for every component slot permitted by the tensor's type, since a partial update, in which only some components are recalculated and others are left in the old basis, produces an array that belongs to no single consistent basis.

Mechanics of the Update

Update Rule for Contravariant Slots

Each contravariant component is updated by summing the old contravariant components multiplied by the corresponding entries of the inverse transformation matrix.

{\bar{T}}^{i} = \sum_{j} {(A^{-1})}_{j}^{i} T^{j}

Update Rule for Covariant Slots

Each covariant component is updated by summing the old covariant components multiplied by the corresponding entries of the transformation matrix itself.

{\bar{T}}_{i} = \sum_{j} A_{i}^{j} T_{j}

Mixed Tensor Update

For a mixed tensor, the update is performed by applying the inverse matrix factor to each contravariant index and the matrix factor to each covariant index, combining all such factors into a single computation that produces each new component from a weighted sum of the old ones.

Order and Independence of Updates

Component-by-Component Computation

Each new component can be computed independently of the others, since the update rule for a given index combination only involves old components sharing compatible index positions and the fixed entries of the transformation matrix, not the values of other new components.

No Dependence on Update Order

Because the update for each new component is computed directly from the old components and the fixed transformation matrix, the order in which the individual new components are computed has no effect on the final result.

Practical Aspects

Matrix Multiplication Implementation

In practice, the component update for tensors of low order is implemented directly as matrix or vector multiplication involving the transformation matrix or its inverse, making the update computationally straightforward once the matrix is known.

Verifying the Update

An update can be checked for correctness by confirming that quantities expected to remain invariant under basis change, such as full contractions yielding scalars, indeed produce the same value when computed from the updated components as when computed from the original ones.