Singular Values of Dual Quaternion Matrices and Their Low-Rank Approximations

The dual quaternion matrix is an important and powerful tool for the study of multi-agent formation control. However, many basic properties of dual quaternion matrices are still not studied, due to the difficulty incurred by the non-commutativity of multiplication and existence of zero divisors of dual quaternion numbers. In this paper, we study some basic properties of dual quaternion matrices, including the polar decomposition theorem, the minimax principle and Weyl's type monotonicity inequality for singular values, spectral norm and the Pythagoras theorem. Based upon these, we further present the best low-rank approximations for dual quaternion matrices, and discuss the relationship between the approximation degree characterizations of the best low-rank approximations in the sense of Frobenius norm and spectral norm, in addition to proving a fundamental property of the best low-rank approximation of dual quaternion matrices in a given subspace. These results are of importance to the applications in rigid body motion and data reduction.