3.14.2 Tensor Covector Inverse Transformation Behavior

Understanding how tensor covectors transform inversely under coordinate changes in tensor algebra.

Tensor Covector Inverse Transformation Behavior is the rule describing how covector components transform when passing back from a new basis to an old one, obtained by reversing the roles of the two bases in the standard covector transformation law and using the inverse of the original change-of-basis matrix in place of the matrix itself. Where the forward transformation carries old covector components to new ones using the matrix A, the inverse transformation carries new components back to old ones using A^{-1}, and examining this reverse direction clarifies why the forward and backward covector rules are mirror images of each other built from the same underlying matrix.

Setting Up the Reverse Direction

The Forward Rule, Restated

Recall that if f_i = A^j_i e_j relates a new basis to an old one, the covector components transform forward according to:

φ_{i}^{'} = \sum_{j = 1}^{n} A_{i}^{j} φ_{j}

Solving for the Old Components

The inverse transformation asks the reverse question: given the new components φ'_i, how are the old components φ_j recovered? Since the forward relation is a linear system with invertible matrix A, it can be solved by multiplying both sides by A^{-1}:

φ_{j} = \sum_{i = 1}^{n} {(A^{- 1})}_{j}^{i} φ_{i}^{'}

This is the inverse transformation behavior: recovering old components from new ones requires the inverse matrix A^{-1}, applied to the primed components.

Symmetry Between Forward and Inverse Rules

The Same Pattern Applied in Reverse

The inverse transformation has exactly the same structural form as the forward transformation, with the roles of old and new swapped and the matrix replaced by its inverse. If the change of basis from the new basis back to the old basis is described by the matrix A^{-1} itself, meaning e_i = (A^{-1})^j_i f_j, then applying the general covector transformation rule to this reversed change of basis reproduces exactly the formula above, confirming that the inverse transformation is simply the forward transformation rule applied to the reversed change of basis.

Consistency Check by Composition

Substituting the inverse transformation into the forward transformation should return the original components unchanged, and this can be verified directly:

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

since the sum over j of A^j_i (A^{-1})^k_j collapses to δ^k_i by the defining property of matrix inverses, leaving only the term with k = i. This confirms the forward and inverse transformations undo each other exactly, as required of any well-defined change of coordinates.

Contrast with the Vector Inverse Transformation

Vector Components Transform Oppositely

Ordinary vector components transform forward using A^{-1} rather than A. It follows that the inverse transformation for vector components, recovering old components from new ones, must use A rather than A^{-1}:

v^{j} = \sum_{i = 1}^{n} A_{i}^{j} v^{i'}

The Crossover Between Forward and Inverse Rules

This produces a notable crossover: the matrix A appears in the forward transformation of covector components and in the inverse transformation of vector components, while A^{-1} appears in the inverse transformation of covector components and in the forward transformation of vector components. Recognizing this crossover is useful for quickly recalling which matrix applies in a given computation, since covariant and contravariant quantities always use opposite matrices for corresponding directions of transformation.

Practical Use of the Inverse Transformation

Recovering Components After a Computation in a New Basis

The inverse transformation behavior is used whenever a computation has been carried out conveniently in a new basis, for instance one aligned with the symmetry of a particular problem, and the result needs to be reported back in terms of the original, standard basis. Applying A^{-1} to the new covector components produces the original components directly, without needing to redo the computation from scratch in the old basis.

Role in Verifying Tensor Well-Definedness

Checking that a proposed object transforms correctly under both the forward and inverse rules is a standard method of confirming that the object is a genuine covector, and not merely an array of numbers that happens to satisfy the forward rule by coincidence in one direction. Since the inverse rule is logically forced by the forward rule together with the invertibility of A, verifying either direction alone is sufficient in principle, but checking both offers a useful consistency safeguard in practice.

Diagrammatic Summary

The diagram shows the two directions of transformation as a round trip between old and new covector components, with the top arrow labeled by A for the forward pass and the bottom arrow labeled by A^{-1} for the inverse pass, together forming a closed loop that returns to the starting components.