Exponential Function - Matrices and Banach Algebras

Matrices and Banach Algebras

The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any Banach algebra B. In this setting, e0 = 1, and ex is invertible with inverse e−x for any x in B. If xy =yx, then ex+y = exey, but this identity can fail for noncommuting x and y.

Some alternative definitions lead to the same function. For instance, ex can be defined as . Or ex can be defined as f(1), where f: R→B is the solution to the differential equation f′(t) = xf(t) with initial condition f(0) = 1.