11.18.4 Tensor Variance Type Transformation Pattern

Understanding how tensor variance types transform under coordinate changes, key to grasping tensor algebra and physical law invariance.

Tensor Variance Type Transformation Pattern is the general template that assembles a tensor's full transformation law from its variance type, applying one inverse basis-change matrix factor for every contravariant index and one direct basis-change matrix factor for every covariant index, all multiplied together and summed over the original indices in a single unified expression.

Foundational Setting

From Type to Law

Once a tensor's variance type is known as the pair $(p, q)$ , the transformation pattern specifies exactly how many matrix factors of each kind must appear, and in what arrangement, to produce the correct transformation law for that specific tensor without needing to rederive it from first principles each time.

The Two Matrix Ingredients

The pattern draws on the basis-change matrix $A$ , relating old and new basis vectors, and its inverse $A^{- 1}$ . Every contravariant index contributes a factor of the inverse, and every covariant index contributes a factor of the direct matrix.

The General Pattern

Full Statement for Type (p, q)

For a tensor of type $(p, q)$ with contravariant indices $i^{1} \dots i^{p}$ and covariant indices $j_{1} \dots j_{q}$ , the transformation pattern reads:

{\tilde{T}}_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} = \sum_{all old indices} (p factors of A inverse) (q factors of A) T_{old lower}^{old upper}

Reading the Pattern in Practice

Each upper index of the new tensor is contracted against exactly one factor of $A^{- 1}$ and one old upper index, while each lower index of the new tensor is contracted against exactly one factor of $A$ and one old lower index, with all contractions carried out inside a single combined summation.

Worked Illustration for a Simple Mixed Case

Type (1, 1) Pattern

For the smallest genuinely mixed case, type $(1, 1)$ , the general pattern specializes to:

{\tilde{T}}_{l}^{k} = \sum_{i, j} {(A^{- 1})}_{i}^{k} A_{l}^{j} T_{j}^{i}

This case already exhibits the complete general pattern in miniature: one inverse factor for the single upper index, one direct factor for the single lower index.

Visual Overview of the Pattern

Diagram of Factor Assignment

Consistency Properties Guaranteed by the Pattern

Composition Under Successive Basis Changes

Because the pattern applies the same matrix factor structure regardless of what specific basis change is used, applying the pattern twice in succession, once for a change from basis one to basis two and again from basis two to basis three, produces the same result as applying the pattern once directly from basis one to basis three, since the matrix products compose correctly at each index independently.

Preservation of Invariants Under the Pattern

The pattern guarantees that whenever a contravariant index of one tensor is contracted against a covariant index of another, or of the same tensor, in forming a product, the corresponding inverse and direct matrix factors introduced by the pattern cancel, leaving the contracted result to follow the pattern appropriate to its own, reduced variance type.

Summary of Key Traits

Defining Characteristics

The transformation pattern assigns one inverse basis-change factor to every contravariant index and one direct basis-change factor to every covariant index.
All factors act within a single combined summation over the tensor's original indices.
The pattern reduces correctly to the pure contravariant, pure covariant, and invariant cases when either count is zero.
The pattern composes consistently under successive basis changes and remains compatible with contraction between tensors.