Linear Systems

Abstract

This chapter presents a study of linear Systems of ordinary differential equations:

$$ {\dot x = }A{\text{x}} $$

(1)

where x ∈ Rⁿ, A is an n × n matrix and

$$ {\dot x}\; = \;\frac{{d{\text{x}}}}{{dt}}\; = \left[ {\begin{array}{*{20}{c}} {\frac{{d{x_1}}}{{dt}}} \\ \vdots \\ {\frac{{d{x_n}}}{{dt}}} \end{array}} \right] $$

It is shown that the solution of the linear System (1) together with the initial condition x(0) = x₀ is given by

$$ {\text{x}}(t)\; = \;{e^{At}}{{\text{x}}_0} $$

where e^At is an n × n matrix function defined by its Taylor series. A good portion of this chapter is concerned with the computation of the matrix e^At in terms of the eigenvalues and eigenvectors of the square matrix A. Throughout this book all vectors will be written as column vectors and A^T will denote the transpose of the matrix A.