fundamental matrix

For a single equation, a fundamental set is a list of independent solution functions. For a system x' = A x, the analog is to line those independent vector solutions up as the columns of one matrix. That matrix, Phi(t), packages every solution at once and is the right object for stating the general solution, applying initial conditions, and even handling forcing.

Concretely, if x_1(t), ..., x_n(t) are n linearly independent solutions of x' = A x, the fundamental matrix is Phi(t) = [ x_1(t) | ... | x_n(t) ], one solution per column. It satisfies the matrix equation Phi'(t) = A Phi(t), and its columns are independent exactly when its determinant (the Wronskian of the system) is nonzero. The general solution is then simply x(t) = Phi(t) c for a constant vector c, and the solution with initial value x(t_0) = x_0 is x(t) = Phi(t) Phi(t_0)^(-1) x_0 — the combination Phi(t) Phi(t_0)^(-1) is the state-transition matrix that propagates any state forward in time.

The fundamental matrix is the central computational object for linear systems and the foundation of linear control and dynamical-systems theory. It also delivers the forced solution cleanly: variation of parameters for systems gives x_p = Phi(t) times integral of Phi(s)^(-1) g(s) ds. When the special normalization Phi(t_0) = I is chosen, the fundamental matrix is exactly the matrix exponential e^(A(t - t_0)) — so e^(At) is just the particular fundamental matrix that equals the identity at t = 0.