Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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Every alternating multilinear matrix function is antisymmetric under a column swap

Statement

Let n≥1n\ge1 and let F:Mn(R)→RF:M_n(R)\to R be column-multilinear and alternating. Interchanging any two columns of AA negates F(A)F(A). This holds over every commutative ring, including characteristic 22.

Facts & Assumptions

Given: An alternating column-multilinear function FF and two selected column positions containing uu and vv.

[L1]

Alternation makes FF vanish on equal columns, while multilinearity expands sums separately in each column (Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring).

[L2]

Ring addition has additive inverses and satisfies the usual sign identities (In any ring 0⋅a=a⋅0=00 \cdot a = a \cdot 0 = 0, (−a)b=a(−b)=−(ab)(-a)b = a(-b) = -(ab), (−a)(−b)=ab(-a)(-b) = ab, (−1)a=−a(-1)a = -a and a(b−c)=ab−aca(b - c) = ab - ac).

Proof

technique · direct
1.1

Put u+vu+v in both selected positions. Alternation gives 0=F(…,u+v,…,u+v,…)0=F(\ldots,u+v,\ldots,u+v,\ldots).

L1
2.1

Expanding twice gives 0=F(…,u,…,v,…)+F(…,v,…,u,…)0=F(\ldots,u,\ldots,v,\ldots)+F(\ldots,v,\ldots,u,\ldots) because the two equal-column terms vanish. Adding the inverse of the first term yields the antisymmetry formula, without dividing by 22.

step 1.1L1L2algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

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