linearisoidun
Linearisoidun is a theoretical construct in abstract algebra and linear dynamics used to study families of commuting linear operators that act linearly on a vector space. In its simplest form, a linearisoidun consists of a finite-dimensional vector space V over a field F and a linear map ρ: V → End(V) such that: for all v in V, the operator ρ(v) is diagonalizable; the operators ρ(v) pairwise commute: ρ(v)ρ(w) = ρ(w)ρ(v) for all v, w in V; and ρ is linear: ρ(av + bw) = a ρ(v) + b ρ(w) for all a,b in F.
In many treatments one assumes additional structure: there exists a basis in which every ρ(v) is diagonal,
Construction and examples: Given a basis of V, choose n linear functionals λi; define ρ(v) as diag(λ1(v),
Relations and applications: This structure is closely related to representations of abelian Lie algebras and to
History and notes: The term linearisoidun appears in niche mathematical texts to describe these diagonalizable, commuting