Fundamental matrix (linear differential equation)

For other senses of the term, see Fundamental matrix (disambiguation).

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 $t$.

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.