The problem of constructing Lyapunov's reducing transformation (Q1359898)

For a linear system \(\dot x= A(t)x\) with a continuous periodic matrix \(A(t)\), Lyapunov's reducing transformation is defined as a matrix \(Q\), such that \(B=Q^{-1} (A(t)Q -\dot Q)\) is a constant matrix. The ansatz \(x=Qy\) transforms the system into \(\dot y=By\). The matrix \(Q\) can be constructed by means of successive replacement of the variables. New conditions for the convergence are given, and the rate of convergence is estimated. The results imply a stability test for the zero equilibrium.