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 ${\displaystyle \Psi (t)}$ is a non-singular matrix for all ${\displaystyle t}$.^[2] Moreover, if the entries of ${\displaystyle A(t)}$ are continuous in ${\displaystyle t}$, any solution to ${\displaystyle {\dot {\Psi }}(t)=A(t)\Psi (t)}$ which is a non-singular matrix for any single value of ${\displaystyle t}$, is automatically a non-singular matrix at all other values of ${\displaystyle t}$. Thus in this case, to check that ${\displaystyle \Psi }$ is a fundamental matrix for this equation, it sufficient to check that it is non-singular at a single value of ${\displaystyle t}$.

Moreover, if there is at-least one choice of fundamental matrix for a given system, then for each choice of non-singular matrix ${\displaystyle B}$, there is exactly one fundamental matrix solution ${\displaystyle \Psi }$ such that ${\displaystyle \Psi (0)=B}$^[3]. The same result holds if ${\displaystyle 0}$ is replaced with any fixed value ${\displaystyle t_{0}}$. Also, if ${\displaystyle \Psi _{0}}$ is any fundamental matrix for this equation, then for any non-singular matrix ${\displaystyle C}$, the matrix ${\displaystyle \Psi (t)=\Psi _{0}(t)C}$ is also a fundamental matrix. In particular, if ${\displaystyle \Psi _{0}}$ is any fixed fundamental solution for a given equation, then all other fundamental solutions for this equation are of the form ${\displaystyle \Psi _{0}(t)C}$.

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.^[4]

See also

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