Definition 7.5.1.
A linear system of differential equations is a system of \(n\) coupled first-order linear differential equations of the form:
\begin{align*}
x_1'(t) \amp = a_{11}x_1(t) + a_{12}x_2(t) + \cdots + a_{1n}x_n(t)\\
x_2'(t) \amp = a_{21}x_1(t) + a_{22}x_2(t) + \cdots + a_{2n}x_n(t)\\
\amp \vdots \\
x_n'(t) \amp = a_{n1}x_1(t) + a_{n2}x_2(t) + \cdots + a_{nn}x_n(t)
\end{align*}
where \(x_1(t), x_2(t), \ldots, x_n(t)\) are unknown functions of the independent variable \(t\text{,}\) and \(a_{ij}\) are constant real coefficients.
This system can be written compactly in matrix form as:
\begin{equation*}
\mathbf{x}'(t) = A\mathbf{x}(t)
\end{equation*}
where the coefficient matrix \(A\) and the vector function \(\mathbf{x}(t)\) are defined as:
\begin{equation*}
A = \begin{bmatrix}
a_{11} \amp a_{12} \amp \cdots \amp a_{1n} \\
a_{21} \amp a_{22} \amp \cdots \amp a_{2n} \\
\vdots \amp \vdots \amp \ddots \amp \vdots \\
a_{n1} \amp a_{n2} \amp \cdots \amp a_{nn}
\end{bmatrix}, \quad
\mathbf{x}(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \\ \vdots \\ x_n(t) \end{bmatrix}, \quad
\mathbf{x}'(t) = \begin{bmatrix} x_1'(t) \\ x_2'(t) \\ \vdots \\ x_n'(t) \end{bmatrix}
\end{equation*}
The system is homogeneous (no external forcing terms) and has constant coefficients (the matrix \(A\) does not depend on \(t\)). Each entry \(a_{ij}\) represents the coefficient of function \(x_j(t)\) in the \(i\)-th differential equation.
