Abstract: Introduces a new multidimensional algebranamed the quad-quaternion. Several matrix operations which are commonly used in array signal processing, including eigenvalue decomposition (EVD) and singular value decomposition (SVD), are extended to quad-quaternions. A low rank approximation algorithm for quad-quaternion matrices is proposed and further applied to vector-sensor array signal modeling and direction of arrival estimation. The quad-quaternion based direction of arrival(DOA) estimation algorithms are demonstrated to be superior to the classical methods with respect to memory requirement, accuracy of subspace estimation, and robustness to model error due to more accurate low rank approximation obtained via quad-quaternion based EVD or SVD.