Fundamental matrix (linear differential equation)

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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

is a matrix-valued function whose columns are linearly independent solutions of the system. 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 .[1]

Control theory[edit]

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.


  1. ^ Chi-Tsong Chen (1998). Linear System Theory and Design (3rd ed.). New York: Oxford University Press,. ISBN 978-0195117776. 

See also[edit]