# Linear Systems

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Part of the book series: Texts in Applied Mathematics ((TAM,volume 7))

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## Abstract

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

$$\dot x = Ax$$
((1))

where xR 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.

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### Cite this chapter

Perko, L. (2001). Linear Systems. In: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol 7. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0003-8_1

• DOI: https://doi.org/10.1007/978-1-4613-0003-8_1

• Publisher Name: Springer, New York, NY

• Print ISBN: 978-1-4612-6526-9

• Online ISBN: 978-1-4613-0003-8

• eBook Packages: Springer Book Archive