expEf

expEF is a term used in linear algebra and numerical analysis to denote the matrix exponential of the product EF, where E and F are square matrices compatible for multiplication. In this usage, expEF is shorthand for exp(EF), with exp(X) defined by the power series exp(X) = sum_{k=0}^∞ X^k / k!.

The calculation of exp(EF) relies on standard methods for the matrix exponential. If EF is diagonalizable, EF = V Λ V^{-1} with Λ diagonal, and exp(EF) = V exp(Λ) V^{-1}, where exp(Λ) is the diagonal matrix of e^{λ_i}. When diagonalization is not available or efficient, numerical approaches such as Padé approximants with scaling and squaring are commonly used. For large-scale problems, Krylov subspace methods may be employed to approximate exp(EF) acting on a vector, avoiding the full matrix exponential.

Applications of exp(EF) arise in solving linear differential equations of the form x'(t) = EF x(t) with initial condition x(0) = x0, where the solution is x(t) = exp(t EF) x0. The exponential map also features in Lie theory, where exp maps elements of a Lie algebra to a corresponding Lie group, linking the matrix exponential to transformations in GL(n).

Notation and usage notes: expEF is not a universally standardized symbol and may appear as exp(EF) or expm(EF) in different texts or software documentation. Readers should confirm the intended meaning in context. See also matrix exponential, exponential map, expm.