In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations

is a matrix-valued function whose columns are linearly independent solutions of the system.[1] Then every solution to the system can be written as , for some constant vector (written as a column vector of height n).

One can show that a matrix-valued function is a fundamental matrix of if and only if and is a non-singular matrix for all .[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


  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.