<p>Full-rank factorization is a matrix factorization form widely applied in high-dimensional data analysis, network flow modeling, and data fitting. Its study is of great significance in high-dimensional algebra, especially for quaternion matrices. This paper proposes a high-precision quaternion full-rank factorization algorithm (QFRF) based on the Gram–Schmidt orthogonalization method. Full-rank factorization is employed to derive the general representations of generalized inverses over the quaternion ring. To demonstrate its practical utility, the proposed algorithm is applied to multi-scale color face data for dimensionality reduction and feature extraction, and further evaluated in image recognition tasks, illustrating the effectiveness of the QFRF framework in real-world scenarios.</p>