The Matrix Exponential and the Variations-of-Constants Formula as Fixed-Point Iterations [Lecture Notes]

The starting point is to rewrite the differential equation as an integral equation, which is then viewed as a functional equation. This makes obvious what the fixed point iteration should be to obtain a solution. In the linear case, the fixed-point iteration is seen to yield the so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Neumann series</i>. The concept of kernel representations of operators as continuum generalizations of matrices is introduced to explicitly calculate the terms of the Neumann series. This yields the matrix exponential function in the time-invariant case without having to “guess” the series definition. In the course of this exposition, various fundamental concepts, such as contractions, asymptotic nilpotence, kernel representations, the Cauchy formula for repeated integration, and operator norm bounds, are encountered. Some of these topics are introduced in several sidebars as they are of interest in their own right. The circle of ideas explored also serves as a gentle introduction to useful tools from functional analysis.