6.17.5 Tensor p q Transformation Pattern

The Tensor p q Transformation Pattern defines how tensors change under coordinate transformations based on their index types and space structure.

Tensor p q Transformation Pattern is the general rule describing how the components of an arbitrary type (p, q) tensor change under a change of basis, combining p independent applications of the inverse transition matrix on the contravariant indices with q independent applications of the forward transition matrix on the covariant indices into a single unified transformation law. This pattern is the ultimate generalization of every specific transformation law encountered for vectors, covectors, and the low-order tensors of type (2,0), (1,1), and (0,2), and it is the formal criterion that determines whether any given indexed array of numbers is a genuine tensor.

Stating the General Pattern

The Full Transformation Formula

Given a change of basis with transition matrix A, where e'_i = A^k_i e_k, and its inverse B = A^{-1}, a type (p, q) tensor transforms as:

T_{' j_{1} \dots j_{q}}^{i_{1} \dots i_{p}} = B_{k_{1}}^{i_{1}} \dots B_{k_{p}}^{i_{p}} A_{j_{1}}^{l_{1}} \dots A_{j_{q}}^{l_{q}} T_{l_{1} \dots l_{q}}^{k_{1} \dots k_{p}}

Reading this formula reveals the pattern precisely: every upper index of the new components is paired with a factor of B contracted against a corresponding old upper index, and every lower index of the new components is paired with a factor of A contracted against a corresponding old lower index, with all of these contractions happening simultaneously and independently.

Recovering Familiar Special Cases

Setting p = 1, q = 0 recovers the vector transformation law v'^i = B^i_k v^k; setting p = 0, q = 1 recovers the covector law φ'_j = A^l_j φ_l; setting p = 1, q = 1 recovers the operator conjugation law T' = B T A; and setting p = 2, q = 0 or p = 0, q = 2 recovers the double-B or double-A laws for the type (2,0) and (0,2) classifications respectively.

Why the Pattern Takes This Specific Form

Consistency with Contraction

The transformation pattern is constructed precisely so that contracting any upper index of a type (p, q) tensor against any lower index produces a tensor whose components transform correctly as a type (p-1, q-1) object. This consistency follows from the identity A^l_k B^k_i = δ^l_i, which arises whenever a summed pair consisting of one A factor and one B factor collapses to the identity matrix, cancelling out and leaving the correct reduced transformation law intact.

Consistency with the Tensor Product

The pattern is also additive under tensor products: if S transforms according to its own type (p_1, q_1) pattern and T according to its type (p_2, q_2) pattern, then S ⊗ T automatically transforms according to the type (p_1+p_2, q_1+q_2) pattern, since the transformation factors for S and for T simply multiply together without interference, one set of factors acting only on the indices originating from S and the other only on those from T.

Verifying an Array Is a Genuine Tensor

The Transformation Pattern as a Test

An array of numbers indexed with p upper and q lower positions is only entitled to be called a type (p, q) tensor if its components, computed in every basis, are related by exactly this transformation pattern; arrays that happen to carry the right number of indices but fail to satisfy the pattern, such as a table of measurements indexed by coincidence rather than by geometric or algebraic necessity, are not tensors despite superficially resembling one.

Basis Independence as the Payoff

The entire purpose of the transformation pattern is to guarantee that operations built from the tensor, such as full contractions to a scalar, remain basis-independent; verifying the pattern once, in the abstract, replaces the need to re-verify basis independence separately for every new operation performed on the tensor.

Diagram of the General Pattern

Practical Consequences of the General Pattern

Predicting Behavior of Composite Expressions

Once the general transformation pattern is understood, the transformation behavior of any composite tensor expression, however many indices it involves, can be predicted immediately by tracking, for each remaining free index after all contractions, whether it originated from an upper or lower slot and applying the corresponding factor of B or A.

Foundation for Invariant Theory

The transformation pattern is the starting point for invariant theory, the study of which combinations of tensor components remain unchanged under all changes of basis; such invariants necessarily arise from full contractions of tensors whose upper and lower index counts match, since only then do all the A and B factors cancel completely, leaving a pure scalar independent of coordinates.