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).

One can show that 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

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

References

1. Somasundaram, D. (2001). "Fundamental Matrix and Its Properties". Ordinary Differential Equations: A First Course. Pangbourne: Alpha Science. pp. 233–240. ISBN   1-84265-069-6.
2. Chi-Tsong Chen (1998). Linear System Theory and Design (3rd ed.). New York: Oxford University Press. ISBN   0-19-511777-8.
3. Kirk, Donald E. (1970). Optimal Control Theory. Englewood Cliffs: Prentice-Hall. pp. 19–20. ISBN   0-13-638098-0.