Linear Ordinary Differential Equations
The most simple example of dynamical system is a system of linear, constant-coefficients ordinary differential equations, i.e. let and :
whose direct closed-form solution involves computation of the matrix exponential:
Another way, provided the solution is restricted to the local Lebesgue space of -dimensional vector fields, is to use its Laplace transform . In this case
The matrix function is called the resolvent matrix of the differential operator . It is meromorphic with respect to the complex parameter since its matrix elements are rational functions whose denominator is equal for all to . Its polar singularities are the eigenvalues of, whose order equals their index for it, i.e. .
Read more about this topic: Jordan Matrix
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