Homogeneous linear system with constant coefficients

Definition

Let $n \ge 1$ and let $A$ be a real $n \times n$ matrix. The homogeneous linear system with constant coefficients given by $A$ is the system of linear differential equations $$x'(t) = A\, x(t)$$ for an unknown differentiable function $x : \mathbb{R} \to \mathbb{R}^n$.

Remarks

Its solutions are $x(t) = e^{t A} x(0)$. See the matrix exponential as the solution. If $A v = \lambda v$, then $x(t) = e^{\lambda t} v$ is a solution. The eigenvalue method builds all solutions from such solutions. Example: for $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, the solutions are rotations: $x(t) = \left(a \cos t + b \sin t, -a \sin t + b \cos t\right)$.

Explanations

L1 human

The rate of change of the vector $x$ is the matrix $A$ applied to $x$, and $A$ does not change with time. This is the vector version of $y' = a y$, whose solutions are $e^{a t} y(0)$.