12.18.5 Tensor Operation Transformation Compatibility

Tensor Operation Transformation Compatibility is the requirement that a tensor operation's coordinate formula, when re-expressed in a different basis via the appropriate change-of-basis transformation, produce the identical abstract result as the original formula computed in the original basis — equivalently, that the operation commute with arbitrary invertible changes of basis on the spaces involved. This is the property that ultimately distinguishes a genuine tensor operation from a coordinate-dependent numerical recipe that merely happens to look reasonable in one particular basis: an operation satisfying transformation compatibility defines a single well-defined map on abstract tensors, while one that fails it defines, at best, a different map for every choice of basis.

Statement of the Compatibility Requirement

Equivariance Under Change of Basis

For a tensor operation φ and an invertible change-of-basis transformation A acting on the coordinates of a tensor T (with the induced action on tensors of the appropriate variance type), transformation compatibility requires that applying φ after re-expressing T in the new basis agree with re-expressing the result of φ(T) in the new basis. This single equation is the precise mathematical content behind the informal statement that "a tensor operation must not depend on which basis was used to write it down."

Contrast with a Merely Coordinate-Wise Recipe

A rule that operates directly on the numerical array of coordinates of T — for instance, "add 1 to every coordinate" — generally fails this equivariance condition, since applying the rule before or after a change of basis produces different results in general. Such a rule fails to define any operation on abstract tensors at all; it is tied to the specific basis in which it was first written down.

Verifying Transformation Compatibility for Standard Operations

Contraction's Compatibility with Change of Basis

The contraction of a mixed tensor's matched contravariant and covariant indices produces a value independent of the basis used to compute it, since a change-of-basis matrix A and its inverse A⁻¹ — acting respectively on the contravariant and covariant indices — cancel exactly in the summed expression. This cancellation is the concrete mechanism by which contraction satisfies transformation compatibility, and it is the reason contraction of a (1,1)-tensor produces a basis-independent scalar (its trace) rather than a basis-dependent number.

Tensor Product's Compatibility

If two tensors transform correctly under a change of basis — meaning each individually satisfies transformation compatibility for the identity operation of "being a tensor" — their tensor product transforms by the corresponding combined action on the enlarged index set, which is exactly what the tensor product operation's own definition produces, confirming the tensor product itself preserves transformation compatibility for any transformation-compatible inputs.

Pullback and Pushforward's Compatibility

Because the pullback and pushforward formulas are defined purely in terms of the abstract linear map f acting on vectors, without reference to any particular basis, expressing both sides of the pullback or pushforward formula in a new basis and comparing produces the same conclusion regardless of which basis was chosen, confirming both operations satisfy transformation compatibility as a direct consequence of being defined coordinate-independently in the first place.

Diagram of Transformation Compatibility as a Commuting Square

Consequences of Failing Transformation Compatibility

The Operation Depends on an Arbitrary Choice

If a proposed operation fails transformation compatibility, its output for a tensor T differs depending on which basis was used to carry out the computation, even though T itself, as an abstract object, has not changed. This makes the operation's result an artifact of an arbitrary bookkeeping choice rather than a genuine property of T, disqualifying it as a legitimate tensor operation regardless of how reasonable its coordinate formula appears in isolation.

Detecting Failure by Testing Two Bases

A practical method for checking transformation compatibility is to compute the operation's result in two different, explicitly related bases and verify the two results correspond correctly under the known change-of-basis transformation; disagreement here is a definitive demonstration that the candidate operation is not basis-independent and therefore not a genuine tensor operation.

Relationship to the Broader Compatibility Framework

The Culminating Requirement Among Compatibility Types

Where variance type compatibility, space compatibility, and basis compatibility govern whether an operation's formula can be evaluated at all for given inputs, transformation compatibility governs something more fundamental: whether the operation, once evaluated correctly in any one basis satisfying the other compatibility requirements, is actually computing a meaningful, basis-independent tensor quantity rather than an artifact tied to that particular basis.

Necessity for Any Operation to Qualify as Genuinely Tensorial

An operation that satisfies every other compatibility requirement — correct variance type, correct ambient space, correctly matched bases for its inputs — but fails transformation compatibility does not qualify as a tensor operation in the full sense used throughout tensor algebra, since the defining feature of tensors and the operations acting on them is precisely this independence from the arbitrary choice of coordinate system.