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Lyapunov functions play a fundamental role in analyzing the stability andconvergence properties of optimization methods. In this paper, we propose anovel and straightforward approach for constructing Lyapunov functions forfirst-order methods applied to quadratic functions. Our approach involvesbringing the iteration matrix to an upper triangular form using Schurdecomposition, then examining the value of the last coordinate of the statevector. This value is multiplied by a magnitude smaller than one at eachiteration. Consequently, this value should decrease at each iteration, providedthat the method converges. We rigorously prove the suitability of this Lyapunovfunction for all first-order methods and derive the necessary conditions forthe proposed function to decrease monotonically. Experiments conducted withgeneral convex functions are also presented, alongside a study on thelimitations of the proposed approach.  Remarkably, the newly discovered Lyapunov function is straightforward anddoes not explicitly depend on the exact method formulation or functioncharacteristics like strong convexity or smoothness constants. In essence, asingle expression serves as a Lyapunov function for several methods, includingHeavy Ball, Nesterov Accelerated Gradient, and Triple Momentum, among others.To the best of our knowledge, this approach has not been previously reported inthe literature.

Another approach to build Lyapunov functions for the first order methods in the quadratic case