# Lyapunov transformation

A non-degenerate linear transformation $L ( t) : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n}$( or $L ( t) : \mathbf C ^ {n} \rightarrow \mathbf C ^ {n}$), smoothly depending on a parameter $t \in \mathbf R$, that satisfies the condition

$$\sup _ {t \in \mathbf R } [ \| L ( t) \| + \| L ^ {-1} ( t) \| + \| \dot{L} ( t) \| ] < + \infty .$$

It was introduced by A.M. Lyapunov in 1892 (see [1]). The Lyapunov transformation is widely used in the theory of linear systems of ordinary differential equations. In many cases the requirement

$$\sup _ {t \in \mathbf R } \| \dot{L} ( t) \| < + \infty$$