Liouville's Formula - Statement of Liouville's Formula

Statement of Liouville's Formula

Consider the n-dimensional first-order homogeneous linear differential equation

on an interval I of the real line, where A(x) for x ∈ I denotes a square matrix of dimension n with real or complex entries. Let Φ denote a matrix-valued solution on I, meaning that each Φ(x) is a square matrix of dimension n with real or complex entries and the derivative satisfies

Let

denote the trace of A(ξ) = (a_i, j(ξ))_{i, j ∈ {1,...,n}}, the sum of its diagonal entries. If the trace of A is a continuous function, then the determinant of Φ satisfies

for all x and x₀ in I.