16.4.3 Tensor Antisymmetric Component Relation

The Tensor Antisymmetric Component Relation describes how antisymmetric parts of tensors behave under index swaps, revealing key properties in multilinear algebra.

Tensor Antisymmetric Component Relation is the set of equations linking the individual scalar components of an antisymmetric tensor to one another once a basis has been fixed, expressing the abstract antisymmetry condition as concrete constraints between numbers indexed by basis labels.

From Abstract Antisymmetry to Component Equations

The Two-Index Case

For a rank-2 antisymmetric tensor T with components T_{ij} relative to a basis, the antisymmetric relation between components reads:

T_{i j} = - T_{j i}

for every pair of basis indices i, j. This single relation, applied to all n² component slots of an n × n array, is what reduces the number of independent components from n² down to n(n − 1)/2.

Diagonal Consequence

Setting i = j in the relation forces every diagonal component to satisfy T_{ii} = −T_{ii}, hence:

T_{i i} = 0 for every i (char ≠ 2)

so the component relation automatically zeroes out the entire diagonal, not just for some special tensor but for every antisymmetric tensor without exception.

Component Relations for Higher Rank

General Rank-k Relation

For an antisymmetric tensor of rank k, the component relation extends to every transposition of index pairs:

T_{i_{1} \dots i_{a} \dots i_{b} \dots i_{k}} = - T_{i_{1} \dots i_{b} \dots i_{a} \dots i_{k}}

for any two positions a, b among the k index slots, with all other indices held in place.

Full Permutation Relation

Combining all pairwise relations yields the general component relation under any permutation σ of the k indices:

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

Vanishing on Repeated Index Labels

Any component with two equal index labels is forced to zero by the same reasoning as the diagonal case:

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

so nonzero components can only occur when all k index labels are pairwise distinct.

Counting Independent Components from the Relation

Reduction Formula

Since the component relation ties every arrangement of a given set of k distinct indices to a single value up to sign, the number of independent components equals the number of ways to choose an unordered set of k distinct indices from n available labels:

# independent components = (\binom{n}{k})

Worked Reduction for Rank 2 in Three Dimensions

In n = 3 dimensions, a rank-2 antisymmetric tensor has C(3,2) = 3 independent components, matching the three independent entries of a 3 × 3 antisymmetric matrix:

T = (\begin{matrix} 0 & T_{12} & T_{13} \\ - T_{12} & 0 & T_{23} \\ - T_{13} & - T_{23} & 0 \end{matrix})

with T_{21}, T_{31}, T_{32} fully determined by the relation as the negatives of T_{12}, T_{13}, T_{23}.

Component Relation Under Basis Change

Transformation Compatibility

The antisymmetric component relation is preserved under any change of basis, since a linear change of coordinates acts on both sides of the relation identically:

T_{i' j'}^{'} = A_{i' i} A_{j' j} T_{i j}

so if T_{ij} = −T_{ji} in the old basis, the same relation T'_{i'j'} = −T'_{j'i'} automatically holds in the new basis, confirming antisymmetry is a basis-independent property even though the individual component values change.

Diagram of the Relation Grid

The diagram shows how three free values (T12, T13, T23) determine all nine grid entries through the component relation, with the diagonal fixed at zero and the lower triangle mirroring the upper triangle with reversed sign.