Linear Systems

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

EquationSource <m:semantics> <m:mrow> <m:mover accent='true'> <m:mi>x</m:mi> <m:mo>&#x02D9;</m:mo> </m:mover> <m:mo>=</m:mo><m:mfrac> <m:mrow> <m:mi>d</m:mi><m:mi>x</m:mi></m:mrow> <m:mrow> <m:mi>d</m:mi><m:mi>t</m:mi></m:mrow> </m:mfrac> <m:mo>=</m:mo><m:mrow><m:mo>[</m:mo> <m:mrow> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:mfrac> <m:mrow> <m:mi>d</m:mi><m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> <m:mrow> <m:mi>d</m:mi><m:mi>t</m:mi></m:mrow> </m:mfrac> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mo>.</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mo>.</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mo>.</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:mfrac> <m:mrow> <m:mi>d</m:mi><m:msub> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> <m:mrow> <m:mi>d</m:mi><m:mi>t</m:mi></m:mrow> </m:mfrac> </m:mrow> </m:mtd> </m:mtr> </m:mtable></m:mrow> <m:mo>]</m:mo></m:mrow></m:mrow> </m:semantics> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ \dot x = \frac{{dx}}{{dt}} = \left[ {\begin{array}{*{20}{c}} {\frac{{d{x_1}}}{{dt}}} \\ . \\ . \\ . \\ {\frac{{d{x_n}}}{{dt}}} \end{array}} \right]$$

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

EquationSource<m:math display='block'> <m:mrow> <m:mi>x</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:msup> <m:mi>e</m:mi> <m:mrow> <m:mi>A</m:mi><m:mi>t</m:mi></m:mrow> </m:msup> <m:msub> <m:mi>x</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ x\left( t \right) = {e^{At}}{x_0}$$

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.