Fundamental matrix (linear differential equation)

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 every solution to the system can be written as $\mathbf {x} =\Psi (t)\mathbf {c}$, for some constant vector $\mathbf {c}$ (written as a column vector 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 $t$.^[1]

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.

References

1. Chi-Tsong Chen (1998). Linear System Theory and Design (3rd ed.). New York: Oxford University Press,. ISBN 978-0195117776.