13.8 Tensor Trace Contraction Case

Tensor Trace Contraction Case is the special instance of tensor contraction in which a single contravariant index and a single covariant index belonging to the same tensor are paired and summed, producing the direct multilinear-algebra generalization of the matrix trace. It identifies the narrow but foundational case where contraction acts on exactly one mixed index pair drawn from one tensor, rather than on multiple pairs, multiple tensors, or indices distributed across separate factors.

Conceptual Basis

Origin in the Matrix Trace

For a linear map represented by a matrix, the trace is obtained by summing the diagonal entries, which in index notation corresponds to setting the row and column indices equal and summing. The tensor trace contraction case extends this same operation to a mixed tensor by contracting its single upper index against its single lower index.

Minimal Contraction Unit

Among all forms of contraction, the trace case represents the smallest nontrivial unit: exactly one contravariant slot paired with exactly one covariant slot, with any additional indices, if present, left untouched as free slots on the resulting object.

Basis Independence of the Trace Value

Because the trace case pairs a contravariant index with a covariant index, the summed result is invariant under a change of basis for a rank-two mixed tensor, since the transformation factors from the Jacobian and its inverse cancel exactly across the paired slot.

Formal Description

Rank-Two Case

For a mixed tensor $$T_j^i$$, the trace contraction case is written:

where the repeated index $$i$$ is summed over its full range, yielding a scalar.

Higher-Rank Case

For a tensor with additional free indices, such as $$T_{ik}^{ij}$$, the trace contraction case applies to only one designated mixed pair, here $$i$$, leaving $$j$$ and $$k$$ free:

producing a partial trace with respect to the chosen index pair.

Dimensional Requirement

The trace contraction case requires that the contracted upper and lower indices range over spaces of identical dimension, which is automatically satisfied when both indices originate from the same underlying vector space, as is typical for endomorphism-type tensors.

Distinguishing Features

Single-Tensor Scope

Unlike contractions that link indices across two or more distinct tensors, the trace contraction case is defined entirely within a single tensor, using only indices that already belong to that one object.

Relation to Eigenvalue Sums

For a rank-two mixed tensor representing a linear endomorphism, the trace contraction case yields a scalar equal to the sum of the eigenvalues of the corresponding linear map, counted with algebraic multiplicity.

Relation to Full and Partial Contraction

When the tensor being traced has no indices beyond the paired pair, the trace contraction case coincides with full contraction, producing a scalar with nothing left over. When additional free indices remain, the trace contraction case is one specific instance of a broader partial contraction applied to that tensor.

Applications

Invariant Extraction From Linear Operators

The trace contraction case is the standard means of extracting a single invariant scalar characterizing a linear operator represented as a mixed tensor, independent of the basis used to express its components.

Building Block for Iterated Traces

Repeated application of the trace contraction case, contracting one mixed pair at a time on tensors formed from products of an operator with itself, produces the sequence of power traces used to characterize the operator's spectral properties.

Use in Physical Field Quantities

In tensor fields such as the stress-energy tensor or the Ricci tensor, the trace contraction case is applied with respect to a metric tensor to produce scalar invariants, such as scalar curvature, that carry coordinate-independent physical meaning.