Alphabeta MathHow statement and proof provenance work
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Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring
Definition
Fix and a commutative ring . A function is column-multilinear when, with all other columns fixed, it is additive and compatible with scalar multiplication in each selected column.
It is alternating when whenever two columns of are equal. It is antisymmetric when interchanging two columns negates its value. It is normalized when . These conditions are stated over the ring itself and do not assume that is invertible.
Depends on
Used by
- Over ℤ/2, an antisymmetric bilinear form need not be alternating Counterexample
- Every alternating multilinear F satisfies F(A)=F(I)∑_σ∈ Sₙsgn(σ)∏ᵢ a_σ(i),i Lemma
- Every alternating multilinear matrix function is antisymmetric under a column swap Lemma
- The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring Theorem
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- S. New, MATH 146 Linear Algebra 1 Lecture Notes, Definition 4.17 (standard reference, not scraped)
- P. Massot, Structures algébriques fondamentales, Definition 6.4.1 (standard reference, not scraped)