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

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).

A matrix-valued function

[2] 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.

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