In mathematics, Liouville's formula, also known as the Abel–Jacobi–Liouville identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system.
Let Φ denote a matrix-valued solution on I, meaning that Φ(t) is the so-called fundamental matrix, a square matrix of dimension n with real or complex entries and the derivative satisfies Let denote the trace of A(s) = (ai, j (s))i, j ∈ {1,...,n}, the sum of its diagonal entries.
This example illustrates how Liouville's formula can help to find the general solution of a first-order system of homogeneous linear differential equations.
Since the trace of A(x) is zero for all x ∈ I, Liouville's formula implies that the determinant is actually a constant independent of x.
It remains to show that this representation of the derivative implies Liouville's formula.
Therefore, g has to be constant on I, because otherwise we would obtain a contradiction to the mean value theorem (applied separately to the real and imaginary part in the complex-valued case).