Exponentials, Differential Equations and One-parameter Subgroups

Abstract

The matrix versions of the familiar real and complex exponential and logarithm functions are fundamental for the study of many aspects of matrix group theory, particularly the one-parameter subgroups. Indeed, the matrix exponential function provides the link between the Lie algebra of a matrix group and the group itself. In the case of a compact connected matrix group, the exponential is even surjective, allowing a parametrisation of such a group by a region in ℝ n for some n; see Chapter 10 for details. Just as in the theory of ordinary differential equations, matrix exponential functions also play a central rôle in the theory of certain types of differential equations for matrix-valued functions and these are important in many applications of Lie theory.