Fundamental matrix (linear differential equation)

In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations$$\dot{𝐱}(t)=A(t)𝐱(t)$$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 $𝐱(t)=\mathrm{\Psi }(t)𝐜$, for some constant vector $𝐜$ (written as a column vector of height Template:Mvar).

A matrix-valued function $\mathrm{\Psi }$ is a fundamental matrix of $\dot{𝐱}(t)=A(t)𝐱(t)$ if and only if $\dot{\mathrm{\Psi }}(t)=A(t)\mathrm{\Psi }(t)$ and $\mathrm{\Psi }$ is a non-singular matrix for all Template:Nowrap

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

References

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