The Lyapunov matrix is a crucial element of the functions and functionals with prescribed derivatives along the solutions of differential equations. This matrix allows to analyze stability both for ordinary differential and time-delay linear time-invariant (LTI) systems. For delay-free systems the stability analysis requires testing for positive definiteness of the Lyapunov matrix. In the case of time-delay systems the Lyapunov matrix allows to construct an auxiliary block-matrix which plays the same role. This paper is devoted to a new formula for the Lyapunov matrix for both delay-free and time-delay systems. The formula is based on the theory of analytic continuation from complex analysis. It is shown how the formula can be applied to some problems related to the Lyapunov-Krasovskii stability approach.