Liouville's formula

The Liouville formula (named after Joseph Liouville ) is an identity which links the determinant of the fundamental matrix of a linear ordinary differential equations of first order with the trace of the coefficient matrix. Using the Liouville formula one can for example easily prove the abelian identity.

Statement

Be an interval, steady and a matrix solution of

Ie is differentiable. Then for all the Liouville formula

Conclusions

• In particular, either for all or for a regular matrix no. In the former case it is called a fundamental matrix solution or shortly fundamental matrix. Applies addition, it means the principal fundamental matrix solution.

• Be a solid matrix. In the special case of the matrix exponential function is obtained from the Liouville formula