Linear Systems

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


This chapter presents a study of linear systems of ordinary differential equations:
$$ \dot{x}=Ax $$
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.


Linear System Phase Portrait Real Eigenvalue Solution Curve Complex Eigenvalue 
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Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Lawrence Perko
    • 1
  1. 1.Department of MathematicsNorthern Arizona UniversityFlagstaffUSA

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