expJ

expJ is a term used in mathematics and related disciplines to denote the exponential of an operator or matrix J, usually written as exp(J) or e^{J}. It is defined for a square matrix or linear operator on a finite-dimensional vector space by the convergent power series e^{J} = ∑_{k=0}^{∞} J^{k}/k!.

In practice, many properties of exp(J) follow from the structure of J. If J is diagonalizable, J = P D P^{-1}, and exp(J) = P exp(D) P^{-1}, where exp(D) is the diagonal matrix with entries e^{λ_i}. For Jordan blocks, the computation uses J = λI + N with N nilpotent, giving exp(J) = e^{λ} ∑_{k=0}^{m-1} N^{k}/k!. The nature of J influences the result: if J is skew-symmetric, exp(J) is orthogonal; if J is symmetric, exp(J) is symmetric and positive definite; if J is nilpotent, exp(J) is a finite polynomial.

expJ has several common applications. It provides solutions to linear differential equations x'(t) = J x(t), with x(t) = exp(J t) x(0). In physics and computer graphics, it describes continuous transformations such as rotations and time evolution. In control theory, exp(J t) serves as the state-transition matrix. Numerically, exp(J) is often computed using scaling and squaring combined with Padé approximants, with care taken for numerical stability and efficiency.

Related concepts include the exponential map in Lie theory, where exp maps a Lie algebra element J to a corresponding Lie group element. See also matrix exponential, Lie group, and rotation matrix.