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MATH
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Definitions from Linear ODE systems

All definitions from Linear ODE systems.

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First-order linear system

A first-order linear system with constant coefficient matrix AA has the form

x′(t)=Ax(t),\mathbf{x}'(t)=A\mathbf{x}(t),

where x(t)=(x1(t),…,xn(t))T\mathbf{x}(t)=(x_1(t),\ldots,x_n(t))^{\mathsf T} and AA is an n×nn\times n matrix. Specifying x(0)=x0\mathbf{x}(0)=\mathbf{x}_0 makes this an initial-value problem.

Writing the equations as one matrix equation records all couplings in AA. For example,

x1′=x1+3x2,x2′=3x1+x2⟺x′=(1331)x.\begin{aligned} x_1'&=x_1+3x_2,\\ x_2'&=3x_1+x_2 \end{aligned} \qquad\Longleftrightarrow\qquad \mathbf{x}'= \begin{pmatrix} 1&3\\ 3&1 \end{pmatrix}\mathbf{x}.
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First-order linear system