The article examines the estimation of Toeplitz covariance matrices by employing overparameterized gradient descent methods. Specifically, it focuses on maximizing the Gaussian log-likelihood function while adhering to Toeplitz constraints, which impose a structured form on the covariance matrices. The study highlights the effectiveness of simple gradient descent techniques in achieving this optimization goal, demonstrating their capability within this specialized context. This approach offers a novel perspective on covariance estimation, particularly relevant given recent advancements in deep learning. The findings support the positive stance that gradient descent methods can be successfully applied to structured covariance estimation problems. Overall, the research contributes to the broader understanding of how classical optimization methods can be adapted and leveraged in modern machine learning frameworks.