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 $$\dot{\mathbf{𝐱}}(t)=A(t)\mathbf{𝐱}(t)$$ is a matrix-valued function $\mathrm{\Psi }(t)$ whose columns are linearly independent solutions of the system. Then every solution to the system can be written as $\mathbf{𝐱}(t)=\mathrm{\Psi }(t)\mathbf{𝐜}$, for some constant vector $\mathbf{𝐜}$ (written as a column vector of height n).

One can show that a matrix-valued function $\mathrm{\Psi }$ is a fundamental matrix of $\dot{\mathbf{𝐱}}(t)=A(t)\mathbf{𝐱}(t)$ if and only if $\dot{\mathrm{\Psi }}(t)=A(t)\mathrm{\Psi }(t)$ and $\mathrm{\Psi }$ 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