expT

expT is a shorthand that is sometimes used to denote the matrix exponential of a square matrix T, written as exp(T) or e^T. While exp(T) is the standard notation in most texts, some authors or software adopt expT as a compact identifier for the same object.

Definition and basic properties: For a square matrix T, the matrix exponential is defined by the convergent power series exp(T) = sum_{k=0}^∞ T^k / k!. Equivalently, if T can be put into Jordan form T = PJP^{-1}, then exp(T) = P exp(J) P^{-1}. If T has eigenvalues λ_i, then exp(T) has eigenvalues exp(λ_i). Key identities include exp(0) = I, and det(exp(T)) = exp(tr(T)). In general exp(A) exp(B) = exp(A + B) if A and B commute.

Time-parameter interpretation and computation: When considered as a function of a scalar t, the map t -> exp(tT) solves the linear differential equation X'(t) = T X(t) with X(0) = I. Numerically, exp(T) is computed via methods such as scaling and squaring combined with Padé approximants, or via eigen decomposition when available.

Applications: The matrix exponential is central in solving linear systems of differential equations, in describing continuous-time Markov chains, in control theory for state transition matrices, and in quantum mechanics for time evolution operators. It also appears in numerical linear algebra and various areas of applied mathematics.

Ambiguity and related terms: Because notation varies, expT should be interpreted from context. Related concepts include the time-ordered exponential in physics and the exponential of a linear operator in functional analysis.