10.12.3 Tensor Transformation Matrix Direction Convention

Explaining how tensor transformation matrices adjust direction under coordinate changes, and the standard conventions used in tensor algebra.

Tensor Transformation Matrix Direction Convention is the explicit agreement fixing whether the transformation matrix is defined to carry the old basis to the new basis or the new basis to the old basis, a choice that must be settled once and adhered to consistently, since the labels forward and inverse, and the corresponding assignment of matrix factors to contravariant and covariant indices, depend entirely on which of these two directions has been designated as the primary one. It exists because the relationship between two bases is symmetric in principle, admitting a matrix in either direction, and only a stated convention resolves which of the two equally valid matrices is meant whenever the unqualified term transformation matrix is used.

The Choice Being Fixed

Old to New as the Primary Direction

Under the most common convention, the transformation matrix is defined so that it expresses the new basis vectors as linear combinations of the old ones, making this direction, from old to new, the primary sense of the term forward matrix.

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

The Opposite Convention

An equally valid but less commonly adopted convention instead defines the primary matrix as the one carrying the new basis vectors back to the old ones, effectively swapping which matrix is called forward and which is called its inverse relative to the first convention.

Consequences of the Chosen Convention

Determining Which Index Type Uses Which Matrix

Once a direction convention is fixed, the entire assignment of forward and inverse matrix factors to contravariant and covariant indices follows automatically; adopting the opposite direction convention would simply exchange the roles, so that contravariant indices would use what was previously called the forward matrix and covariant indices would use what was previously called the inverse matrix.

v^{i}' = {(A^{- 1})}_{j}^{i} v^{j}

No Effect on the Underlying Physics or Geometry

Because the two possible direction conventions differ only by which matrix is labeled forward and which is labeled inverse, adopting one convention over the other changes no invariant fact about how a tensor or a change of basis behaves; only the notation and the direction in which formulas are read change.

Necessity of Stating the Convention Explicitly

Preventing Silent Ambiguity

Without an explicit statement of the direction convention being used, a reader cannot determine, from the symbol alone, whether a given matrix carries old-to-new or new-to-old, making it impossible to correctly assign matrix factors to contravariant and covariant indices without guessing.

Maintaining a Single Convention Throughout a Body of Work

Once a direction convention has been adopted at the start of a discussion, it must be maintained consistently throughout every subsequent formula, since switching conventions partway through would silently invert the meaning of every matrix factor from that point onward, without any visible notational cue that the switch had occurred.

Schematic Representation

The diagram contrasts the two possible directions the transformation matrix could be defined to point, illustrating why an explicit convention must be stated before any formula involving forward and inverse matrices can be interpreted unambiguously.