10.6 Tensor Inverse Basis Change Rule

Tensor Inverse Basis Change Rule is the transformation law describing how tensor components and basis vectors must be updated when moving from a new basis back to the original basis, using the matrix inverse to that governing the forward change. It exists as the necessary counterpart to the forward change of basis: whenever a forward rule specifies how a new basis is built from an old one, the inverse rule specifies the reverse construction, and consistency between the two is what allows a change of basis to be undone without loss of information.

Motivation for the Rule

Reversibility of Basis Change

A change of basis is only meaningful if it can be reversed. If the new basis vectors are expressed as linear combinations of the old basis vectors through a coefficient matrix, that matrix must be invertible, and the inverse basis change rule is precisely the statement of how the old basis and old components are recovered from the new ones using the inverse matrix.

Existence Condition

The inverse basis change rule can only be formulated when the forward coefficient matrix is nonsingular. This existence condition is not an extra assumption imposed on top of tensor algebra; it is already required for the forward change of basis to define a genuine basis in the first place, since a set of vectors obtained through a singular linear combination of a basis would fail to be linearly independent.

Component Transformation Under the Inverse Rule

Contravariant Components

Contravariant components, which transform with the inverse of the forward matrix when moving to the new basis, transform with the forward matrix itself when the inverse rule is applied to return to the old basis.

Covariant Components

Covariant components follow the opposite pattern: having transformed with the forward matrix to reach the new basis, they transform with the inverse matrix under the inverse basis change rule to return to the old basis.

General Mixed Tensors

For a tensor with several upper and lower indices, the inverse basis change rule applies one factor of the forward matrix to every upper index and one factor of the inverse matrix to every lower index, mirroring exactly the opposite assignment used by the forward rule.

Structural Properties

Cancellation Under Composition

Applying the forward basis change rule followed immediately by the inverse basis change rule must return every basis vector and every component to its original value. This cancellation property is a direct consequence of the defining relation between a matrix and its inverse, and it serves as a consistency check on any explicit computation of a change of basis.

Symmetry With the Forward Rule

The inverse basis change rule is not an independently chosen rule but is completely determined once the forward rule and its coefficient matrix are fixed. This symmetry means that no additional information beyond the forward matrix is required to state the inverse rule, only the operation of matrix inversion.

Preservation of Tensor Invariance

Just as the forward rule preserves the invariant meaning of a tensor when moving to a new basis, the inverse rule preserves that same invariant meaning when moving back. A tensor reconstructed from old-basis components and old basis vectors after a full forward-then-inverse cycle is identical to the tensor before the cycle began.

Schematic Representation

The upper arrow represents the forward basis change rule carrying the old basis to the new basis, while the lower arrow represents the inverse basis change rule carrying the new basis back to the old one, with the two coefficient matrices being exact inverses of each other.