Fundamental matrix (linear differential equation)

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

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

One can show that a matrix-valued function Ψ is a fundamental matrix of x ˙ ( t ) = A ( t ) x ( t ) if and only if Ψ ˙ ( t ) = A ( t ) Ψ ( t ) and Ψ is a non-singular matrix for all t .