|
|
Pure Mathematics 2026
关于对偶复矩阵的笛卡尔分解的酉不变范数不等式
|
Abstract:
研究了对偶复矩阵的笛卡尔分解的酉不变范数不等式。给出了对偶复矩阵的笛卡尔分解的定义。利用对偶向量间的优超关系以及对偶复矩阵的Mirsky定理证明了对偶复矩阵的笛卡尔分解的一个酉不变范数不等式。该不等式揭示了一个对偶复矩阵与其笛卡尔分解中两个对偶Hermite复矩阵的特征值的酉不变范数的数量关系。
Unitarily norm inequalities for the Cartesian decomposition of dual complex matrices are studied. The definition of Cartesian decomposition of a dual complex matrix is defined. By using the majorization relation between dual vectors and Mirsky theorem of dual complex matrices, a unitarily norm inequality for the Cartesian decomposition of dual complex matrices is proved, which reveals the quantity relation between the unitarily norm of a dual complex matrix and the two dual Hermitian complex matrices in its Cartesian decomposition.
| [1] | Clifford, (1871) Preliminary Sketch of Biquaternions. Proceedings of the London Mathematical Society, 1, 381-395. https://doi.org/10.1112/plms/s1-4.1.381 |
| [2] | Perez, A. and McCarthy, J.M. (2003) Dual Quaternion Synthesis of Constrained Robotic Systems. Journal of Mechanical Design, 126, 425-435. https://doi.org/10.1115/1.1737378 |
| [3] | Valverde, A. and Tsiotras, P. (2018) Spacecraft Robot Kinematics Using Dual Quaternions. Robotics, 7, Article 64. https://doi.org/10.3390/robotics7040064 |
| [4] | Angeles, J. (1998) The Application of Dual Algebra to Kinematic Analysis. In: Angeles, J. and Zakhariev, E., Eds., Computational Methods in Mechanical Systems, Springer, 3-32. https://doi.org/10.1007/978-3-662-03729-4_1 |
| [5] | Condurache, D. (2019) A Davenport Dual Angles Approach for Minimal Parameterization of the Rigid Body Displacement and Motion. Mechanism and Machine Theory, 140, 104-122. https://doi.org/10.1016/j.mechmachtheory.2019.05.011 |
| [6] | Leclercq, G., Lefèvre, P. and Blohm, G. (2013) 3D Kinematics Using Dual Quaternions: Theory and Applications in Neuroscience. Frontiers in Behavioral Neuroscience, 7, Article 7. https://doi.org/10.3389/fnbeh.2013.00007 |
| [7] | Abaunza, H., Castillo, P., Lozano, R. and Victorino, A. (2016) Quadrotor Aerial Manipulator Based on Dual Quaternions. 2016 International Conference on Unmanned Aircraft Systems (ICUAS), Arlington, 7-10 June 2016, 152-161. https://doi.org/10.1109/icuas.2016.7502589 |
| [8] | Valverde, A. and Tsiotras, P. (2018) Dual Quaternion Framework for Modeling of Spacecraft-Mounted Multibody Robotic Systems. Frontiers in Robotics and AI, 5, Article 128. https://doi.org/10.3389/frobt.2018.00128 |
| [9] | Qiao, B., Tang, S., Ma, K. and Liu, Z. (2013) Relative Position and Attitude Estimation of Spacecrafts Based on Dual Quaternion for Rendezvous and Docking. Acta Astronautica, 91, 237-244. https://doi.org/10.1016/j.actaastro.2013.06.022 |
| [10] | Yang, A.T. and Freudenstein, F. (1964) Application of Dual-Number Quaternion Algebra to the Analysis of Spatial Mechanisms. Journal of Applied Mechanics, 31, 300-308. https://doi.org/10.1115/1.3629601 |
| [11] | Yang, A.T. (1967) Application of Dual Quaternions to the Study of Gyrodynamics. Journal of Engineering for Industry, 89, 137-143. https://doi.org/10.1115/1.3609985 |
| [12] | Yacob, F. and Semere, D. (2020) Variation Compensation in Machining Processes Using Dual Quaternions. Procedia CIRP, 93, 879-884. https://doi.org/10.1016/j.procir.2020.04.034 |
| [13] | Li, G., Zhang, F., Fu, Y. and Wang, S. (2019) Kinematic Calibration of Serial Robot Using Dual Quaternions. Industrial Robot: The International Journal of Robotics Research and Application, 46, 247-258. https://doi.org/10.1108/ir-10-2018-0221 |
| [14] | Luo, J., Chen, S., Zhang, C., Chen, C. and Yang, G. (2023) Efficient Kinematic Calibration for Articulated Robot Based on Unit Dual Quaternion. IEEE Transactions on Industrial Informatics, 19, 11898-11909. https://doi.org/10.1109/tii.2023.3254666 |
| [15] | Luo, J., Chen, S., Jiang, D., Zheng, T., Li, H., Fang, Z., et al. (2024) Efficient Kinematic Calibration for Parallel Manipulators Based on Unit Dual Quaternion. IEEE Transactions on Industrial Informatics, 20, 6791-6801. https://doi.org/10.1109/tii.2024.3353914 |
| [16] | Qi, L.Q. and Luo, Z.Y. (2023) Eigenvalues and Singular Values of Dual Quaternion Matrices. Pacific Journal of Optimization, 19, 257-272. |
| [17] | Ling, C., Qi, L. and Yan, H. (2023) Minimax Principle for Eigenvalues of Dual Quaternion Hermitian Matrices and Generalized Inverses of Dual Quaternion Matrices. Numerical Functional Analysis and Optimization, 44, 1371-1394. https://doi.org/10.1080/01630563.2023.2254090 |
| [18] | Ding, W., Li, Y., Wang, T. and Wei, M. (2024) Dual Quaternion Singular Value Decomposition Based on Bidiagonalization to a Dual Number Matrix Using Dual Quaternion Householder Transformations. Applied Mathematics Letters, 152, Article ID: 109021. https://doi.org/10.1016/j.aml.2024.109021 |
| [19] | Ding, W., Li, Y. and Wei, M. (2025) A New Structure-Preserving Method for Dual Quaternion Hermitian Eigenvalue Problems. Applied Mathematics Letters, 163, Article ID: 109432. https://doi.org/10.1016/j.aml.2024.109432 |
| [20] | Cui, C. and Qi, L. (2024) A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix. Journal of Scientific Computing, 100, Article No. 21. https://doi.org/10.1007/s10915-024-02561-x |
| [21] | Ling, C., Pan, C. and Qi, L. (2025) A Metric Function for Dual Quaternion Matrices and Related Least-Squares Problems. Linear and Multilinear Algebra, 73, 3056-3079. https://doi.org/10.1080/03081087.2025.2491645 |
| [22] | Ling, C., He, H. and Qi, L. (2022) Singular Values of Dual Quaternion Matrices and Their Low-Rank Approximations. Numerical Functional Analysis and Optimization, 43, 1423-1458. https://doi.org/10.1080/01630563.2022.2108835 |
| [23] | Chen, L., Hongjin, H., Liqun, Q. and Tingting, F. (2024) Spectral Norm and Von Neumann Type Trace Inequality for Dual Quaternion Matrices. Pacific Journal of Optimization, 2024, 229-247. https://doi.org/10.61208/pjo-2023-045 |
| [24] | Zhu, L., Wang, Q. and Kou, Z. (2025) The Least-Norm Solution to a Matrix Equation over the Dual Quaterion Algebra. Symmetry, 17, Article 267. https://doi.org/10.3390/sym17020267 |
| [25] | Shi, L., Wang, Q., Xie, L. and Zhang, X. (2024) Solving the Dual Generalized Commutative Quaternion Matrix Equation AXB = C. Symmetry, 16, Article 1359. https://doi.org/10.3390/sym16101359 |
| [26] | Chen, Y., Wang, Q. and Xie, L. (2024) Dual Quaternion Matrix Equation AXB = C with Applications. Symmetry, 16, 287. https://doi.org/10.3390/sym16030287 |
| [27] | Cheng, S. and Hu, H.F. (2024) Unitarily Invariant Norms on Dual Quaternion Matrices. Pacific Journal of Optimization, 2024, 371-387. https://doi.org/10.61208/pjo-2023-051 |
| [28] | Zhan, X. (1999) Norm Inequalities for Cartesian Decompositions. Linear Algebra and its Applications, 286, 297-301. https://doi.org/10.1016/s0024-3795(98)10174-x |
| [29] | Qi, L., Ling, C. and Yan, H. (2022) Dual Quaternions and Dual Quaternion Vectors. Communications on Applied Mathematics and Computation, 4, 1494-1508. https://doi.org/10.1007/s42967-022-00189-y |
| [30] | Zhong, J. and Zhong, P. (2025) A Mirsky-Type Unitarily Invariant Norm Inequality for Dual Quaternion Matrices and Its Applications. Symmetry, 17, Article 1355. https://doi.org/10.3390/sym17081355 |
