Fundamental matrix (linear differential equation)

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For other senses of the term, see Fundamental matrix, a disambiguation page.

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

 \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t)

is a matrix-valued function  \Psi(t) whose columns are linearly independent solutions of the system. Then the general solution to the system can be written as \mathbf{x} = \Psi(t) \mathbf{c}, where \mathbf{c} ranges over constant vectors (written as column vectors of height n).

One can show that a matrix-valued function  \Psi is a fundamental matrix of  \dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) if and only if  \dot{\Psi}(t) = A(t) \Psi(t) and  \Psi is a non-singular matrix for all  t .[1]


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