16.7 Tensor Zero Repeated Argument Rule
The Tensor Zero Repeated Argument Rule sets a tensor with a repeated index to zero, simplifying tensor expressions in physics and math.
Tensor Zero Repeated Argument Rule is the foundational law stating that an alternating tensor evaluates to zero whenever two of its arguments are equal, standing as one of two equivalent ways (alongside the sign-reversal law) to define what it means for a tensor to be alternating, and serving as the source of nearly every structural consequence attributed to alternating tensors.
Statement of the Rule
The Core Equation
For an alternating tensor T of rank k, if any two of its k arguments coincide, the output is exactly zero:
with the repeated vector v occupying any two of the k argument positions, and the remaining positions filled arbitrarily.
Adoption as a Primary Definition
In many treatments, this rule is taken as the primary defining property of an alternating tensor, with the sign-reversal law derived from it, rather than the reverse; this ordering emphasizes that the zero-on-repetition behavior is the more fundamental geometric fact — signed area or volume genuinely collapses to zero when directions coincide — from which the algebraic sign bookkeeping follows.
Deriving Sign Reversal from the Rule
The Standard Derivation
Starting from the zero repeated argument rule alone, substitute v = u + w into two argument slots and expand using multilinearity:
The first and last terms vanish by the rule applied directly to u and to w, leaving T(...,u,...,w,...) = −T(...,w,...,u,...), the sign-reversal law.
Consequences That Follow from the Rule
Rank Ceiling
If a tensor of rank k is evaluated on any k vectors drawn from an n-dimensional space with k > n, some two of the vectors must be linearly dependent (a stronger form of "repeated"), and extending the rule to linear dependence (via the derived multilinear expansion) forces the tensor to vanish identically once k exceeds n.
Determinant Vanishing on Singular Matrices
Applied to the determinant, viewed as the unique top-rank alternating tensor of its argument columns, the rule directly explains why a matrix with a repeated column has zero determinant, and by extension (through the linear-dependence generalization) why any singular matrix has zero determinant.
Independent Component Reduction
The rule is also the mechanism by which the number of independent components of a rank-k alternating tensor is reduced from n^k to C(n,k): every component with a repeated index is forced to zero by the rule, leaving only components indexed by sets of distinct values.
The Rule versus Its Generalization
From Literal Repetition to Linear Dependence
The rule as stated applies to literal equality of two arguments, but a stronger consequence — vanishing on any linearly dependent set of arguments — follows once multilinearity is invoked to expand a dependent combination back down to a case covered by the literal rule. This generalization is what makes the rule powerful enough to characterize matrix singularity rather than just repeated columns.
Characteristic-2 Caveat
Where the Rule Is Strictly Stronger
In fields of characteristic 2, the zero repeated argument rule is a strictly stronger condition than the sign-reversal law, since the derivation of sign-reversal from the rule still works, but the reverse derivation (rule from sign-reversal) fails, as noted in the terminology boundary between "antisymmetric" and "alternating." In this setting, the rule as stated here is the correct, stronger notion that should be used whenever a tensor is required to be genuinely alternating.