Fundamentalmatrix

Fundamental matrix, in the context of linear dynamical systems, refers to a matrix-valued function that encodes the evolution of solutions to a linear system. In its common form, it is called the fundamental matrix Φ(t) and is defined for the system x'(t) = A(t) x(t), where A(t) is an n×n matrix possibly depending on time. A fundamental matrix is an invertible matrix-valued function that satisfies Φ'(t) = A(t) Φ(t) for all t in an interval, together with the initial condition Φ(t0) = I for some t0. For any initial condition x(t0) = x0, the solution is x(t) = Φ(t) x0, so Φ(t) serves as a state-transition matrix.

In the special case that A(t) = A is constant, the fundamental matrix reduces to the matrix exponential Φ(t) = e^{A (t−t0)}. Then the solution is x(t) = e^{A (t−t0)} x0, and Φ(t) has the semigroup property Φ(t) Φ(s) = Φ(t+s) when t0 = 0.

When A(t) varies with time, Φ(t, t0) is still defined by Φ'(t, t0) = A(t) Φ(t, t0) with Φ(t0, t0) = I, and it yields x(t) = Φ(t, t0) x0. In general, Φ(t, t0) is invertible and Φ(t2, t0) = Φ(t2, t1) Φ(t1, t0).

The fundamental matrix is central in solving linear systems, performing stability analysis, and studying the state transition in control theory. Computing the matrix exponential is a common task, with methods such as series expansion, diagonalization, Jordan form, and numerical schemes like scaling and squaring with Padé approximants. For time-varying A(t), the solution may be expressed using a time-ordered exponential or Magnus expansion.

In other branches of applied mathematics, the term fundamental matrix also appears in Markov chain theory; for absorbing chains, the fundamental matrix N = (I − Q)^{-1} gives expected times to absorption and the expected number of visits to transient states.