To mathematically analyze the convergence of the momentum method on a scalar quadratic function $f(x) = \frac{\lambda}{2} x^2$, the update equations for both the position $x$ and velocity $v$ can be formulated as a coupled system: $\begin{bmatrix} v_{t+1} \ x_{t+1} \end{bmatrix} = \begin{bmatrix} \beta & \lambda \ -\eta \beta & (1 - \eta \lambda) \end{bmatrix} \begin{bmatrix} v_{t} \ x_{t} \end{bmatrix}$. The convergence behavior is entirely governed by the eigenvalues of this $2 imes 2$ transition matrix. Mathematical analysis of this matrix shows that the velocity converges when the hyperparameters satisfy $0 < \eta \lambda < 2 + 2 \beta$. This feasible range is substantially larger than the $0 < \eta \lambda < 2$ constraint required for standard gradient descent, mathematically confirming that large momentum coefficients ($\beta$) safely permit much larger learning rates without divergence.