Fundamental matrix (linear differential equation)

In mathematics, the fundamental matrix of a system of linear differential equations

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

is the matrix $\Psi$ such that the n columns are linearly independent solutions of ${\dot {x}}(t)=A(t)x(t)$.

By definition

${\dot {\Psi }}(t)=A(t)\Psi (t)$

i.e. $\Psi$ is a fundamental matrix of ${\dot {x}}(t)=A(t)x(t)$ if and only if ${\dot {\Psi }}(t)=A(t)\Psi (t)$ and $\Psi$ is a non-singular matrix for all $t$.

See also

• Fundamental matrix (computer vision)
• Fundamental matrix (linear differential equation)
• Fundamental matrix (absorbing markov chain)