Expmualambda

Expmualambda is a term used in mathematical and computational contexts to denote a parametric exponential map applied to a linear operator, combining two scalar or matrix parameters often denoted μ and λ. It is not a standard term with a unique universal definition; different sources may adopt different conventions.

A common interpretation treats expmualambda as the matrix exponential of a scaled operator: expmualambda(A, μ, λ) = exp((μ λ) A), where A is a square matrix and μ, λ are scalars. Another frequent convention introduces two operators A and B and defines expmualambda(A, B; μ, λ) = exp(μ A + λ B). In specialized contexts, μ and λ may themselves be matrices, yielding expmualambda(A, μ, λ) = exp(μ A λ) or exp(μ A + λ B), depending on how the dependence is modeled.

Computation follows standard matrix-exponential techniques: scaling and squaring with a Padé approximation, or eigenvalue decompositions when applicable. Numerical stability depends on the spectrum of the underlying operator and on the conditioning of the scaling.

Applications include solving linear differential equations with time-dependent or parameter-dependent operators, modeling continuous-time Markov processes with parametric generators, and control theory where μ and λ adjust system dynamics.

Because expmualambda is not a universally defined term, readers should consult the specific source for the intended interpretation in that context. See also matrix exponential, expm, scaling and squaring, Padé approximation, and Lie groups.