13.14.5 Tensor Contraction Residual Tensor Type

Tensor Contraction Residual Tensor Type refers to the resulting tensor after contracting indices, preserving structural properties in algebraic operations.

Tensor Contraction Residual Tensor Type is the classification of the tensor object that remains after a contraction operation removes one contravariant index and one covariant index from a mixed tensor. It describes not the numerical outcome of the contraction but the algebraic category the outcome belongs to: its remaining rank, its distribution of upper and lower indices, and the vector space (or tensor product of spaces) in which it lives.

Definition

Given a tensor of type $(p, q)$ , meaning it has $p$ contravariant indices and $q$ covariant indices, a single contraction pairs one upper index with one lower index and sums over the shared range. The residual tensor type is the type of the object that results from this operation.

T \in V_{q}^{p} \to C (T) \in V_{q - 1}^{p - 1}

Here $V_{q}^{p}$ denotes the space of tensors of type $(p, q)$ over a fixed vector space $V$ , and $C$ is the contraction map. The residual type is therefore always of the form $(p - 1, q - 1)$ : rank drops by exactly one in each variance class.

Rank Reduction

Total Order

Contraction reduces total tensor order by two, since one upper and one lower index are removed simultaneously.

order (C (T)) = order (T) - 2

Variance Balance

The contravariant count and covariant count each decrease by exactly one, preserving the difference $p - q$ that characterizes the tensor's variance signature. A type-(2,1) tensor contracted once yields a type-(1,0) tensor; a type-(1,1) tensor contracted once yields a type-(0,0) scalar.

Index Position Independence

The residual type depends only on how many upper and lower indices are consumed, not on which specific index positions were chosen. Contracting the first upper index with the second lower index produces a residual tensor of the same type as contracting the second upper index with the first lower index, although the resulting component values generally differ.

Structural Consequences

Component Space

The components of the residual tensor occupy an array of dimension $n^{p + q - 2}$ when the underlying vector space has dimension $n$ , compared to $n^{p + q}$ for the original tensor.

Repeated Contraction

Applying contraction repeatedly to a tensor of type $(p, q)$ yields a sequence of residual types:

(p, q) \to (p - 1, q - 1) \to (p - 2, q - 2) \to \dots

terminating when either the upper or lower index count reaches zero, at which point no further contraction is possible without introducing a metric to raise or lower indices.

Full Contraction

When $p = q$ , iterated contraction can proceed until the residual type is $(0, 0)$ , at which point the residual is a scalar, invariant under coordinate transformation.

Diagrammatic View

Invariance of Type Under Basis Change

Because contraction is defined intrinsically via the natural pairing between a vector space and its dual, the residual tensor type does not depend on the choice of basis or coordinate system. A change of basis alters the components of the residual tensor according to the standard tensor transformation law for type $(p - 1, q - 1)$ objects, but never changes the type itself.