Liouville's formula

The Liouville formula (named after Joseph Liouville (1809–1882)) is an identity that links the determinant of the fundamental matrix of a linear system of ordinary differential equations of the first order with the trace of the coefficient matrix. With the help of Liouville's formula, for example, one can easily prove the Abelian identity .

statement

Let be an interval, continuous and a matrix solution of ${\displaystyle J\subset \mathbb {R} }$ ${\displaystyle A\colon J\rightarrow \mathbb {R} ^{n\times n}}$ ${\displaystyle \Phi }$

${\displaystyle \ y'(x)=A(x)y(x)\ ,}$

that is, is differentiable with . Then Liouville's formula applies to all of them ${\displaystyle \Phi \colon J\rightarrow \mathbb {R} ^{n\times n}}$ ${\displaystyle \Phi '(x)=A(x)\Phi (x)}$ ${\displaystyle x,\,x_{0}\in J}$

${\displaystyle \det \Phi (x)=\det \Phi (x_{0})\cdot \exp \left(\int _{x_{0}}^{x}{\rm {Spur}}(A(\xi )){\rm {d}}\xi \right)\ .}$

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 . ${\displaystyle \Phi (x)}$ ${\displaystyle x\in J}$ ${\displaystyle x\in J}$ ${\displaystyle \Phi }$ ${\displaystyle \Phi (x_{0})=I}$ ${\displaystyle \Phi }$ ${\displaystyle x_{0}}$
• Be a solid matrix. In the special case of the matrix exponential function , one obtains from the Liouville formula ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ ${\displaystyle \Phi (x):=e^{xA}}$

${\displaystyle \det(e^{xA})=e^{x\cdot {\textrm {Spur}}(A)}\ ,}$

since is the main fundamental matrix solution for in . ${\displaystyle \Phi }$ ${\displaystyle y'(x)=Ay(x)}$ ${\displaystyle 0}$

literature

• Carmen Chicone: Ordinary Differential Equations with Applications. 2nd Edition. ( Texts in Applied Mathematics , 34) Springer-Verlag, 2006, ISBN 0-387-30769-9 .

Web links

Wikibooks: Proof of Liouville's Formula