This letter proposes an efficient method for calculating the eigenvalues of large-scale exponential correlation matrices by leveraging tridiagonal matrix theory. The approach explicitly factorizes the characteristic polynomial into two lower-degree polynomials, preserving the distinction between odd and even matrix orders without resorting to approximations. The ability to efficiently compute these eigenvalues is critical for optimizing channel capacity, transmit beamforming, and other key operations in massive multiple-input multiple-output (MIMO) systems, which are essential for improving data rates, reliability, and spectral efficiency in wireless communications. This work derives the approximate eigenvalues using the Cauchy interlacing theorem. It then validates the accuracy of the proposed eigenvalue expressions by comparing them with existing expressions in the literature and applying them to massive MIMO capacity estimation, showing improved precision.