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 formula_2 whose columns are linearly independent solutions of the system. Then every solution to the system can be written as formula_3, for some constant vector formula_4 (written as a column vector of height "n"). One can show that a matrix-valued function formula_5 is a fundamental matrix of formula_1 if and only if formula_7 and formula_5 is a non-singular matrix for all formula_9. 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.