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 $\mathbf {x} (t)=\Psi (t)\mathbf {c}$, for some constant vector $\mathbf {c}$ (written as a column vector of height n).

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.

See also

• Linear differential equation
• Liouville's formula
• Systems of ordinary differential equations