  Lyapunov functions play a fundamental role in analyzing the stability and
convergence properties of optimization methods. In this paper, we propose a
novel and straightforward approach for constructing Lyapunov functions for
first-order methods applied to quadratic functions. Our approach involves
bringing the iteration matrix to an upper triangular form using Schur
decomposition, then examining the value of the last coordinate of the state
vector. This value is multiplied by a magnitude smaller than one at each
iteration. Consequently, this value should decrease at each iteration, provided
that the method converges. We rigorously prove the suitability of this Lyapunov
function for all first-order methods and derive the necessary conditions for
the proposed function to decrease monotonically. Experiments conducted with
general convex functions are also presented, alongside a study on the
limitations of the proposed approach.
  Remarkably, the newly discovered Lyapunov function is straightforward and
does not explicitly depend on the exact method formulation or function
characteristics like strong convexity or smoothness constants. In essence, a
single expression serves as a Lyapunov function for several methods, including
Heavy Ball, Nesterov Accelerated Gradient, and Triple Momentum, among others.
To the best of our knowledge, this approach has not been previously reported in
the literature.
