expAB

expAB denotes the matrix exponential of the product AB, where A and B are square matrices of compatible size. If AB is an n by n matrix, the matrix exponential exp(AB) is defined by the power series exp(AB) = I + AB + (AB)^2/2! + (AB)^3/3! + ... . This makes exp(AB) well-defined for any AB whose spectrum lies in the complex plane, with convergence guaranteed by the properties of the exponential series.

In general, there is no simple relation that reduces exp(AB) to a combination of exp(A) and exp(B). For example, exp(AB) does not equal exp(A) exp(B) in most cases. Special care is needed when A and B commute or when AB has particular structure. If AB is nilpotent (for instance, (AB)^k = 0 for some k), the exponential reduces to a finite polynomial: exp(AB) = I + AB + (AB)^2/2! + ... + (AB)^{k-1}/(k-1)!. This can simplify calculations significantly in such cases.

Computation of exp(AB) is a common task in linear differential equations, control theory, and quantum mechanics. For numerical evaluation, methods such as scaling and squaring combined with Padé approximants are widely used, along with eigenvalue decompositions when AB is diagonalizable. The matrix exponential encodes continuous-time evolution in systems described by AB, with applications ranging from solving x'(t) = AB x(t) to modeling linear dynamical systems and time evolution in physics.