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, so ρ(v) = diag(λ1(v), ..., λn(v)) for some linear functionals λi in V*. In that case the study reduces to the geometry of the n-tuple of linear forms (λ1, ..., λn).

Construction and examples: Given a basis of V, choose n linear functionals λi; define ρ(v) as diag(λ1(v), ..., λn(v)). This yields a linearisoidun. The rank of ρ is the dimension of the subspace spanned by {λi}. If the λi are linearly independent, ρ is injective and encodes distinct coordinate-wise responses to v.

Relations and applications: This structure is closely related to representations of abelian Lie algebras and to simultaneous diagonalization problems. It provides a convenient framework for modeling systems where a vector input selects a diagonal action on a fixed coordinate basis.

History and notes: The term linearisoidun appears in niche mathematical texts to describe these diagonalizable, commuting operator families and is not a widely standardized term across disciplines.