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Linear time-varying (LTV) systems naturally arise when one linearizes nonlinear systems about a trajectory. In contrast the linear time-invariant (LTI) cases which have been thoroughly understood in the analysisand synthesis technologies, many features of the LTV systems are still limited and not clear. This paper addressesthe problems of solution and stability of a general unforced LTV differential state space system. Unlike most ofthe work based on the Lyapunov theory, numerical simulations, or specific constraint systems, the paper proposesthe spectral decompositions of the LTV systems by employing extended eigenpairs and with simple mathematicalderivation. The spectral decompositions reveal the mechanisms of inherent characterization in general LTVsystems, rather than a particular class. Moreover, a novel set of auxiliary equations is developed for guiding andobtaining the extended eigenpairs of its system matrix which completely characterize the LTV systems. The solutionsto perform the commutative systems and the second-order systems with companion form are straightforward. The proposed innovative thinking provides a novel guided way to analyze the LTV systems. These findings areeasily extended to LTI cases. Examples from the literature demonstrate the effectiveness and the superiority of theproposed approaches when compared with other methods. The proposed results may be of great interest in both forscientific research and application.
