# Linear Systems

• Lawrence Perko
Part of the Texts in Applied Mathematics book series (TAM, volume 7)

## Abstract

This chapter presents a study of linear systems of ordinary differential equations:
$$\dot{x}=Ax$$
(1)
where x ∈ R n , A is an n × n matrix and
$$\dot{x}=\frac{{dx}}{{dt}}=\left[{\begin{array}{*{20}{c}} {\frac{{d{x_1}}}{{dt}}} \\ \vdots \\ {\frac{{d{x_n}}}{{dt}}} \\ \end{array}} \right]$$
. It is shown that the solution of the linear system (1) together with the initial condition x(0)=x0 is given by
$$x(t)={e^{{At}}}{x_0}$$
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.

## Keywords

Linear System Phase Portrait Real Eigenvalue Solution Curve Complex Eigenvalue
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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