Let be an interval, continuous and a matrix solution of
that is, is differentiable with . Then Liouville's formula
applies to all of them
Inferences
In particular, there is either a regular matrix for all or none . In the first case one calls a fundamental matrix solution or in short fundamental matrix . If it is also true , the main fundamental matrix solution is called in .
Be a solid matrix. In the special case of the matrix exponential function , one obtains from the Liouville formula
since is the main fundamental matrix solution for in .
literature
Carmen Chicone: Ordinary Differential Equations with Applications. 2nd Edition. ( Texts in Applied Mathematics , 34) Springer-Verlag, 2006, ISBN 0-387-30769-9 .