Fundamental matrix

Definition

Let $I$ be an open interval and let $A : I \to \mathbb{R}^{n \times n}$ be continuous. A fundamental matrix of the homogeneous linear system $x' = A(t) x$ is a differentiable function $\Phi : I \to \mathbb{R}^{n \times n}$ with these two properties. For each $t \in I$, $\Phi'(t) = A(t)\, \Phi(t)$. For each $t \in I$, the matrix $\Phi(t)$ is invertible.

Remarks

Equivalently, the columns of $\Phi$ are $n$ solutions that form a basis of the space of solutions. Each solution is $x(t) = \Phi(t)\, c$ for a unique $c \in \mathbb{R}^n$, and the solution with $x(t_0) = x_0$ is $\Phi(t) \Phi(t_0)^{-1} x_0$. If $\Phi'(t) = A(t) \Phi(t)$ and $\Phi(t_0)$ is invertible at one point $t_0$, then $\Phi(t)$ is invertible at each point, by Liouville's formula. For a constant matrix $A$, $\Phi(t) = e^{tA}$ is a fundamental matrix. See the matrix exponential as the solution.

Explanations

L1 human

A fundamental matrix puts $n$ independent solutions side by side as columns. Then one matrix product gives each solution: choose the vector $c$ of coefficients.