An operator is a function, often one that maps the input object to something else in its input space, such that the same operator can be applied repeatedly to indefinitely transform an object in some "direction".
ββββββββββββββ OPERATOR TOPOLOGY EXPLORER ββββββββββββββββββ
β Visualizing Transformational Functions β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β β
β Basic Operator Schema: β
β βββββββββββββββββββββββββββββββββββββββββββββββ β
β β Input Space β β
β β βββββ Γ βββββ Γ βββββ β β
β β β x βββββββββββΆβx'βββββββββββΆβx''β β β
β β βββββ βββββ βββββ β β
β β β² β β β
β β βββββββββββββββββββββββββββββββ β β
β βββββββββββββββββββββββββββββββββββββββββββββββ β
β β
β Time Evolution Operator: β
β βββββββββββββββββββββββββββββββββββββββββββββββ β
β β TΜ β β
β β tβ βββββββΆ tβ βββββββΆ tβ βββββββΆ tβ β β
β β β β β β β β
β β βΌ βΌ βΌ βΌ β β
β β Ο(tβ) ββββΆ Ο(tβ) ββββΆ Ο(tβ) ββββΆ Ο(tβ) β β
β βββββββββββββββββββββββββββββββββββββββββββββββ β
β β
β Escalation Operator: β
β βββββββββββββββββββββββββββββββββββββββββββββββ β
β β Γ β β
β β βββββ β β
β β β xββ β β
β β βββββ΄ββββ β β
β β β xβ β β
β β βββββ΄ββββ β β
β β β xβ β β β
β β βββββββββ β β
β βββββββββββββββββββββββββββββββββββββββββββββββ β
β β
β Eigenspace Visualization: β
β βββββββββββββββββββββββββββββββββββββββββββββββ β
β β Γ β β
β β β Ξ»βvβ β β
β β β β β β
β β β β Ξ»βvβ β β
β β ββ β β β
β ββββββββββββββββββΆ β β
β β ββ β β β
β β β β Ξ»βvβ β β
β β β β β β
β β βΌ Ξ»βvβ β β
β βββββββββββββββββββββββββββββββββββββββββββββββ β
β β
β Properties: β
β β’ Preservation of structure under transformation β
β β’ Iterative application possible β
β β’ Maps input space back to itself β
β β’ May have fixed points or eigenvectors β
β β
β [Calculate Eigenvalues] [View Transform] [Iterate] β
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ