10.17.3 Tensor Active Transformation Component Effect

Tensor Active Transformation Component Effect explains how components change under active transformations, key in understanding tensor behavior in physics and mathematics.

Tensor Active Transformation Component Effect is the specific numerical change each tensor component undergoes, in the single fixed basis used throughout, as a direct consequence of the underlying point map's action, describing how the new component values relate to the original ones through the transformation matrix built from the map's differential, and how this relationship depends on the variance type of each index exactly as it does under a coordinate change.

Effect on a Contravariant Component

Direct Application of the Transformation Matrix

A contravariant component picks up a new value formed by summing the original components weighted by the transformation matrix built from the active map's differential:

V^{' i} = \sum_{k}^{n} A_{k}^{i} V^{k}

If the map rotates space by a fixed angle, the effect on the components is the same numerical mixing pattern produced by a rotation matrix, and if the map stretches space along a particular direction, the effect is a corresponding numerical increase in the component aligned with that direction.

Mixing Between Directions

Just as in the coordinate transformation case, if the transformation matrix has non-zero off-diagonal entries, the new value of one component genuinely depends on the original values of other components as well, reflecting real mixing of directions caused by the active map, such as the mixing produced by a rotation that is not aligned with any single coordinate axis.

Effect on a Covariant Component

Inverse Transpose Relationship

A covariant component transforms using the inverse of the transformation matrix, applied with indices reversed relative to the contravariant case, to preserve pairings with contravariant vectors under the map action:

W_{' i} = \sum_{k}^{n} {(A^{- 1})}_{i}^{k} W_{k}

For an orthogonal transformation such as a rotation, this inverse equals the transpose of the original matrix, so the covariant component effect for a rotation is described by the same rotation matrix used for the contravariant case, only applied in transposed form.

Effect on Higher-Rank Tensor Components

Compounded Per-Index Effect

A rank-two tensor's components change with one factor of the transformation matrix, or its inverse, per index, exactly mirroring the compounding seen in coordinate transformations:

T^{' ij} = \sum_{k}^{n} \sum_{l}^{n} A_{k}^{i} A_{l}^{j} T^{kl}

Effect on Symmetry and Eigenstructure

For a symmetric rank-two tensor, an active rotation genuinely changes the orientation of the tensor's principal axes relative to the fixed basis while its eigenvalues remain unchanged, since eigenvalues are invariant under an orthogonal similarity transformation, illustrating that the component effect can alter some structural features of a tensor, such as its orientation, while leaving others, such as its eigenvalues, untouched.

Diagram of Component Effect

Numerical Mixing Under a Fixed Basis

Special Component Effects

Zero Components Under an Active Map

A component that vanishes before the transformation need not remain zero afterward, since the active map can introduce a non-zero contribution to that direction from other original components, exactly as in the coordinate transformation case, except here the change reflects a genuine alteration of the tensor rather than a mere change of description.

Fixed Directions of the Map

If a particular direction is left invariant by the active transformation, meaning the transformation matrix has that direction as an eigenvector with eigenvalue one, the component along that specific direction is unaffected by the component effect, even though other components may still change, illustrating that the component effect can act selectively on different parts of a tensor depending on the specific structure of the transformation matrix.

Verifying a Component Effect Computation

Checking Preserved Invariants

For a transformation matrix known to preserve a particular invariant, such as vector length under an orthogonal matrix, computing the length of the transformed vector from its new components and confirming it matches the original length provides a direct check that the component effect has been computed correctly, analogous to the invariance check used to verify a passive coordinate transformation.