10.6.4 Tensor Inverse Basis Component Effect

The Tensor Inverse Basis Component Effect explores how inverse basis components transform tensor properties within algebraic structures.

Tensor Inverse Basis Component Effect is the specific change produced in the numerical values of a tensor's components when the inverse basis change rule is applied, describing how each component of every index type is recomputed from the source frame values through contraction with the inverse coefficient matrix or the forward coefficient matrix, depending on whether that index is contravariant or covariant. It isolates the outcome on the component array itself, separate from the accompanying change in the basis vectors, so that the numerical consequences of reversing a change of basis can be examined directly.

Nature of the Effect

Component-Level Focus

While the inverse basis change rule as a whole governs both basis vectors and components together, the inverse basis component effect refers specifically to what happens to the numbers stored in a tensor's component array. This distinction matters because the transformation experienced by components is not identical to the transformation experienced by basis vectors, even though both are governed by the same underlying coefficient matrix.

Dependence on Index Type

The effect on a contravariant component differs from the effect on a covariant component. A contravariant component recovers its old-basis value by contracting with the forward matrix, while a covariant component recovers its old-basis value by contracting with the inverse matrix, so the component effect cannot be described by a single formula independent of index type.

v^{i} = A_{j}^{i} v^{j}'

ω_{i} = {(A^{- 1})}_{j}^{i} ω_{j}'

Effect on Different Tensor Ranks

Scalars

For a scalar, which carries no free index, the inverse basis component effect is trivial: the single component is unchanged by any change of basis, forward or inverse, since no index remains to be contracted with a coefficient matrix.

Vectors and Covectors

For a vector or a covector, the inverse basis component effect involves exactly one contraction, with the forward matrix in the vector case and the inverse matrix in the covector case, producing a single new set of components from the single set given in the source frame.

Higher-Rank Tensors

For a tensor of higher rank, the inverse basis component effect involves one contraction per index, with each upper index contributing a factor of the forward matrix and each lower index contributing a factor of the inverse matrix, all applied simultaneously to produce the full new component array.

T_{j}^{i} = A_{k}^{i} {(A^{- 1})}_{j}^{l} T_{l}^{k}'

Consequences of the Effect

Numerical Recovery of Original Values

The most direct consequence of the inverse basis component effect is that, when applied to components already produced by the forward rule, it exactly recovers the original component values, since the two effects are inverse operations of one another by construction.

Sensitivity to Matrix Conditioning

Because the inverse basis component effect depends on the inverse coefficient matrix, any numerical sensitivity present in that inverse, for instance when the forward matrix is close to singular, is inherited directly by the recovered components, even though the underlying tensor being represented has not changed.

Independence From the Physical or Geometric Object

Although the component effect can produce substantially different numbers depending on the bases involved, the effect never alters the invariant tensor itself. The inverse basis component effect is purely a statement about how the representation changes, not about any change in the object being represented.

Schematic Representation

The diagram shows two distinct component arrays connected by the inverse basis component effect, both representing the identical invariant tensor despite holding different numerical entries.