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

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]

See also

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