Liouville's formula


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. The formula is named after the French mathematician Joseph Liouville. Jacobi's formula provides another representation of the same mathematical relationship.
Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other, see the example application below.

Statement of Liouville's formula

Consider the -dimensional first-order homogeneous linear differential equation
on an interval of the real line, where for denotes a square matrix of dimension with real or complex entries. Let denote a matrix-valued solution on, meaning that each is a square matrix of dimension with real or complex entries and the derivative satisfies
Let
denote the trace of, the sum of its diagonal entries. If the trace of is a continuous function, then the determinant of satisfies
for all and in.

Example application

This example illustrates how Liouville's formula can help to find the general solution of a first-order system of homogeneous linear differential equations. Consider
on the open interval. Assume that the easy solution
is already found. Let
denote another solution, then
is a square-matrix-valued solution of the above differential equation. Since the trace of is zero for all, Liouville's formula implies that the determinant
is actually a constant independent of. Writing down the first component of the differential equation for, we obtain using that
Therefore, by integration, we see that
involving the natural logarithm and the constant of integration. Solving equation for and substituting for gives
which is the general solution for. With the special choice and we recover the easy solution we started with, the choice and yields a linearly independent solution. Therefore,
is a so-called fundamental solution of the system.

Proof of Liouville's formula

We omit the argument for brevity. By the Leibniz formula for determinants, the derivative of the determinant of can be calculated by differentiating one row at a time and taking the sum, i.e.
Since the matrix-valued solution satisfies the equation, we have for every entry of the matrix
or for the entire row
When we subtract from the  th row the linear combination
of all the other rows, then the value of the determinant remains unchanged, hence
for every by the linearity of the determinant with respect to every row. Hence
by and the definition of the trace. It remains to show that this representation of the derivative implies Liouville's formula.
Fix. Since the trace of is assumed to be continuous function on, it is bounded on every closed and bounded subinterval of and therefore integrable, hence
is a well defined function. Differentiating both sides, using the product rule, the chain rule, the derivative of the exponential function and the fundamental theorem of calculus, we obtain
due to the derivative in . Therefore, has to be constant on, because otherwise we would obtain a contradiction to the mean value theorem. Since, Liouville's formula follows by solving the definition of for.