Department of Mathematics

Dual quaternions form a non-commutative ring widely used in 3D kinematics, robotics, and multi-agent control. However, computing fundamental matrix decompositions, such as the Singular Value Decomposition (SVD), directly over this ring is computationally expensive due to quaternion arithmetic.

In this talk, we will discuss a fast, structure-preserving algorithm for the Dual Quaternion Singular Value Decomposition (DQSVD). By introducing a novel \( \mathcal{L}_{\mathcal{D}} \) representation, this approach maps dual quaternion matrices into real block matrices, effectively converting all operations into pure real arithmetic. This allows the use of real Householder and Givens transformations to bidiagonalize the matrix while strictly preserving its underlying algebraic structure.

We will explore how this method reduces storage costs by a factor of 8, maintains high numerical stability, and achieves a \( 3\times{} \) speedup over classical algorithms. A brief application to color image watermarking will also be reviewed.