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 $\mathrm{\Psi }(t)$ is a non-singular matrix for all Template:Nowrap Moreover, if the entries of $A(t)$ are continuous in $t$, any solution to $\dot{\mathrm{\Psi }}(t)=A(t)\mathrm{\Psi }(t)$ which is a non-singular matrix for any single value of $t$, is automatically a non-singular matrix at all other values of $t$. Thus in this case, to check that $\mathrm{\Psi }$ is a fundamental matrix for this equation, it sufficient to check that it is non-singular at a single value of $t$.

Moreover, if there is at-least one choice of fundamental matrix for a given system, then for each choice of non-singular matrix $B$, there is exactly one fundamental matrix solution $\mathrm{\Psi }$ such that $\mathrm{\Psi }(0)=B$. The same result holds if $0$ is replaced with any fixed value $t_0$. Also, if ${\mathrm{\Psi }}_0$ is any fundamental matrix for this equation, then for any non-singular matrix $C$, the matrix $\mathrm{\Psi }(t)={\mathrm{\Psi }}_0(t)C$ is also a fundamental matrix. In particular, if ${\mathrm{\Psi }}_0$ is any fixed fundamental solution for a given equation, then all other fundamental solutions for this equation are of the form ${\mathrm{\Psi }}_0(t)C$.

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

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

References

Template:Matrix-stub