16.4.2 Tensor Antisymmetric Sign Change Relation

The Tensor Antisymmetric Sign Change Relation describes how swapping indices changes the sign of antisymmetric tensors, a fundamental property in algebraic structures.

Tensor Antisymmetric Sign Change Relation is the equation stating precisely how much and in what direction an antisymmetric tensor's value changes when its indices or arguments are reordered, serving as the master relation from which all other antisymmetry properties of the tensor are derived.

The Core Relation

Statement for a Pair of Indices

For an antisymmetric tensor T, the sign change relation between any two indices i and j states:

T_{\dots i \dots j \dots} = - T_{\dots j \dots i \dots}

This relation is unconditional: it holds for every pair of index positions in the tensor and for every choice of the values those indices take, with no exceptional cases beyond what follows logically from the relation itself.

Multiplicative Nature of the Sign Change

The relation is multiplicative rather than additive: reordering multiplies the tensor's value by −1, it does not add or subtract a correction term:

T_{ji} = (- 1) \times T_{ij}

so the sign change relation is exact, not approximate, and preserves magnitude information entirely.

Iterating the Sign Change Relation

Two Applications Cancel

Applying the sign change relation twice in a row, i.e. reordering and then reordering back, returns the original value, since (−1) × (−1) = 1:

T_{ij} \to - T_{ji} \to (- 1) (- 1) T_{ij} = T_{ij}

General Permutation Extension

For a rank-k antisymmetric tensor, applying the pairwise relation repeatedly to build up an arbitrary permutation σ of the k indices yields:

T_{i_{σ (1)} \dots i_{σ (k)}} = sgn (σ) T_{i_{1} \dots i_{k}}

and the total sign change accumulated is the signature of σ, regardless of which sequence of pairwise swaps was used to reach it.

Consequence: The Vanishing Relation

Derivation from Self-Consistency

Setting i = j inside the core relation gives T_{...i...i...} = −T_{...i...i...}, an equation only satisfiable in fields of characteristic other than 2 if:

T_{\dots i \dots i \dots} = 0

This vanishing relation is not a separate axiom but a direct algebraic consequence of the sign change relation applied to the degenerate case of two equal indices.

Rank Bound Consequence

Combining the vanishing relation with the requirement that indices range over only n basis directions shows that any antisymmetric tensor of rank k > n must be identically zero, since no assignment of k distinct values to the indices is possible once k exceeds n.

Sign Change Relation in Matrix Form

Rank-2 Matrix Representation

Written as a matrix T with entries T_{ij}, the sign change relation is exactly the defining condition of an antisymmetric (skew-symmetric) matrix:

T^{⊤} = - T

where Tᵀ is the matrix transpose. This compact matrix statement is equivalent, entry by entry, to the index-based sign change relation.

Eigenvalue Consequence

A structural consequence of the matrix form is that all real eigenvalues of an antisymmetric matrix must be zero, and any nonzero eigenvalues occur strictly in purely imaginary conjugate pairs, a fact traceable directly to the relation Tᵀ = −T combined with the reality of the matrix entries.

Diagram of the Sign Flip

The diagram shows the sign change relation as a single directional operation: swapping the labeled slots multiplies the tensor's value by −1, and this is the entire content of the relation for any pair of indices, at any rank, in any dimension.