# Fundamental matrix (linear differential equation)

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

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

is a matrix-valued function ${\displaystyle \Psi (t)}$ whose columns are linearly independent solutions of the system.[1] Then every solution to the system can be written as ${\displaystyle \mathbf {x} (t)=\Psi (t)\mathbf {c} }$, for some constant vector ${\displaystyle \mathbf {c} }$ (written as a column vector of height n).

One can show that a matrix-valued function ${\displaystyle \Psi }$ is a fundamental matrix of ${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)}$ if and only if ${\displaystyle {\dot {\Psi }}(t)=A(t)\Psi (t)}$ and ${\displaystyle \Psi }$ is a non-singular matrix for all ${\displaystyle t}$.[2]

## Control theory

The fundamental matrix is used to express the state-transition matrix, an essential component in the solution of a system of linear ordinary differential equations.[3]