Quaternion Matrix Algebra and Applications

Quaternion matrix algebra extends classical matrix theory into the realm of hypercomplex numbers, enabling elegant representation and manipulation of three-dimensional rotations and rigid-body motions. A quaternion matrix consists of entries that are quaternions, a four-dimensional extension of complex numbers featuring one real and three imaginary units. Noncommutativity of quaternion multiplication gives rise to rich algebraic structure and unique challenges in defining determinants, eigenvalues and matrix equations. Dual quaternions further enrich this framework by pairing quaternions with dual numbers, seamlessly encoding rotation and translation in a single entity. Over the past decade, theoretical advances have deepened understanding of solvability conditions for quaternion matrix equations, generalised inverses, and Hermitian properties, while computational innovations have delivered efficient algorithms for eigenanalysis, interpolation and system identification. Applications span computer graphics, robotics, aerospace engineering, signal processing and neuromechanics, where quaternion algebra underpins smooth rotational trajectories, accurate rigid-body localisation and compact representation of high-dimensional transformations.

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Quaternion Matrix Algebra and Applications publication trend

The graph below shows the total number of articles in quaternion matrix algebra and applications across all publications each year (not limited to Nature Index journals).

Technical terms

Quaternion: Hypercomplex number system with one real and three imaginary components, widely used to represent spatial rotations without singularities.

Dual quaternion: Algebraic construct combining quaternions with dual numbers to encode both rotation and translation in rigid-body transformations.

Sylvester-like quaternion matrix equation: Noncommutative linear equation of form AX + XB = C over quaternions, requiring specialised techniques for existence and general solution.

Hermitian quaternion matrix: Quaternion matrix equal to its conjugate transpose, ensuring real eigenvalues and orthogonal eigenspaces under appropriate definitions.

General solution: Complete parameterisation of all matrices satisfying a given quaternionic matrix equation under consistency conditions.

Power method: Iterative algorithm for approximating the dominant eigenvalue and eigenvector of a matrix, here extended to dual quaternion Hermitian matrices.

References

1. Three Symmetrical Systems of Coupled Sylvester-like Quaternion Matrix Equations. Symmetry (2022).
2. Dual Quaternion Matrix Equation AXB = C with Applications. Symmetry (2024).
3. A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix. Journal of Scientific Computing (2024).