One-Parameter Groups of Linear Transformations

A one-parameter linear group in a vector space L is a continuous map $$ g:t\mapsto {g^t}=g(t) $$ on the space of parameters ℝ, taking values in a space of the linear operators on L and having the following properties: (1) g 0 = E and (2) the sum of parameter values corresponds to the composition $$ {g^{s+t }}={g^s}\circ {g^t} $$ . Thus, one-parameter linear groups in ℝ are the continuous solutions of the functional equations g(0) = 1, g(t + s) = g(t)g(s). As is well known, exponential functions satisfy these equations. After solving the first problems in this chapter, readers will find that the continuous solutions are all exponential and, thus, will extract the usual properties of exponential functions from these functional equations (which corresponds to a real historical development). The second part of the chapter is devoted to proper generalizations for one-parameter matrix groups. Readers will find that all of them are given by the matrix exponential series, as in the one-dimensional case. However, a multidimensional picture looks much more complicated. Working on these problems will familiarize readers with matrix analysis. In addition, readers may need familiarity with some tools used by number theory and the theory of differential equations, such as Euclid’s algorithm, Chinese remainder theorem, and Poincaré’s recurrence theorem; we introduce and discuss these tools. In the third part of the chapter readers will find some elementary classic applications of one-parameter matrix groups in differential equation theory (Liouville formula), in complex analysis (e.g., complex exponential functions, Euler’s formula), for finite-dimensional functional spaces (spaces of quasipolynomials), and others. Readers will see how to deal with these problems using elementary tools of analysis and linear algebra. Further, part of this chapter contains problems for readers possessing more advanced preliminary experience (although we introduce and discuss the necessary tools). While working on these problems, readers will come across many interesting things regarding analysis and linear algebra and become acquainted with important concepts (symplectic forms and others). Readers will encounter far-reaching advances and applications of the subjects considered in the present chapters in powerful mathematical theories: of differential equations, Lie groups, and group representations (a guide to the literature is provided).