In this paper, we first show a class relation between the eigenvalue of functional matrix derivative and the derivative of function matrix eigenvalue. Applying the relation, we transform the time-varying linear matrix differential equation into eigenvalue differential equation. Furthermore, by using singular value decomposition and majorisation inequalities, we derive upper and lower bounds on eigenvalue summation of the solution for the Lyapunov matrix differential equation, which improve the recent results. As an application in control and optimisation, we show that our bounds could be used to discuss the stability of a class time-varying nonlinear system. Finally, we give a corresponding numerical example to show the superiority and effectiveness of the derived bounds.