# Fundamental matrix (linear differential equation)

Short description: Matrix consisting of linearly independent solutions to a 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]