Fundamental matrix (linear differential equation)
In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations
[math]\displaystyle{ \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) }[/math]
is a matrix-valued function [math]\displaystyle{ \Psi(t) }[/math] whose columns are linearly independent solutions of the system.[1]
Then every solution to the system can be written as [math]\displaystyle{ \mathbf{x}(t) = \Psi(t) \mathbf{c} }[/math], for some constant vector [math]\displaystyle{ \mathbf{c} }[/math] (written as a column vector of height n).
One can show that a matrix-valued function [math]\displaystyle{ \Psi }[/math] is a fundamental matrix of [math]\displaystyle{ \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) }[/math] if and only if [math]\displaystyle{ \dot{\Psi}(t) = A(t) \Psi(t) }[/math] and [math]\displaystyle{ \Psi }[/math] is a non-singular matrix for all [math]\displaystyle{ t }[/math].[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
References
- ↑ 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. https://books.google.com/books?id=PduY2CjJ1zEC&pg=PA233.
- ↑ Chi-Tsong Chen (1998). Linear System Theory and Design (3rd ed.). New York: Oxford University Press. ISBN 0-19-511777-8.
- ↑ Kirk, Donald E. (1970). Optimal Control Theory. Englewood Cliffs: Prentice-Hall. pp. 19–20. ISBN 0-13-638098-0. https://books.google.com/books?id=onuH0PnZwV4C&pg=PA19.
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