Fundamental matrix (linear differential equation)

For other senses of the term, see Fundamental matrix, a disambiguation page.

In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations

${\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)$

is a matrix-valued function $\Psi (t)$ whose columns are linearly independent solutions of the system. Then the general solution to the system can be written as $\mathbf {x} =\Psi (t)\mathbf {c}$, where $\mathbf {c}$ ranges over constant vectors (written as column vectors of height n).

One can show that a matrix-valued function $\Psi$ is a fundamental matrix of ${\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)$ if and only if ${\dot {\Psi }}(t)=A(t)\Psi (t)$ and $\Psi$ is a non-singular matrix for all $t$.