  This paper presents a systematic theoretical framework to derive the energy
identities of general implicit and explicit Runge--Kutta (RK) methods for
linear seminegative systems. It generalizes the stability analysis of explicit
RK methods in [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal., 57 (2019), pp.
1158-1182]. The established energy identities provide a precise
characterization on whether and how the energy dissipates in the RK
discretization, thereby leading to weak and strong stability criteria of RK
methods. Furthermore, we discover a unified energy identity for all the
diagonal Pade approximations, based on an analytical Cholesky type
decomposition of a class of symmetric matrices. The structure of the matrices
is very complicated, rendering the discovery of the unified energy identity and
the proof of the decomposition highly challenging. Our proofs involve the
construction of technical combinatorial identities and novel techniques from
the theory of hypergeometric series. Our framework is motivated by a discrete
analogue of integration by parts technique and a series expansion of the
continuous energy law. In some special cases, our analyses establish a close
connection between the continuous and discrete energy laws, enhancing our
understanding of their intrinsic mechanisms. Several specific examples of
implicit methods are given to illustrate the discrete energy laws. A few
numerical examples further confirm the theoretical properties.
