expkB

expkB is a mathematical shorthand denoting the matrix exponential of k times a square matrix B, written as exp(kB) or e^(kB). It is a central object in linear algebra, differential equations, and theoretical physics, where B often represents a linear operator or a rate/generator matrix.

Definition and computation: exp(kB) is defined by the power series exp(kB) = sum_{n=0}^∞ (k^n B^n)/n!. If B is diagonalizable as B = P D P^{-1}, then exp(kB) = P exp(kD) P^{-1}, with exp(kD) = diag(exp(k d_i)). For matrices that are not diagonalizable, one uses Jordan form, and the exponential of a Jordan block yields a combination of exponentials and polynomials in k. Practical computation typically employs scaling and squaring with Padé approximants or specialized algorithms.

Key properties: exp(0) = I, and the derivative with respect to k satisfies d/dk exp(kB) = B exp(kB) = exp(kB) B, since exp(kB) is a polynomial (and power series) in B. Determinant and trace have structure linked to B, e.g., det(exp(kB)) = exp(k tr(B)).

Applications: In solving linear systems dx/dt = Bx, the solution is x(t) = exp(tB) x(0). In physics, exp(-iHt/ħ) describes quantum time evolution for a Hamiltonian H. In stochastic processes, exp(tQ) gives the transition dynamics of a continuous-time Markov chain with generator Q. In Lie theory, the exponential map links Lie algebras to Lie groups.

See also: matrix exponential, Jordan form, scaling and squaring, Padé approximants.