Dual Quaternion Generalized Power Method

• DQGPM is an iterative eigensolver that extends classical power methods to dual quaternion matrices for computing dominant eigenvalues and eigenvectors.
• It employs per-iteration projection onto the unit dual quaternion group and normalization techniques to ensure convergence in both Hermitian and non-Hermitian cases.
• Applications in SLAM, SE(3) synchronization, and multi-agent control demonstrate its practical efficacy with rapid convergence and robust error stability.

The Dual Quaternion Generalized Power Method (DQGPM) is a class of eigensolvers designed for the computation of dominant eigenvalues and eigenvectors of dual quaternion matrices, with rigorous theoretical guarantees and practical efficacy in applied geometry, multi-agent systems, and pose estimation on SO(3) or SE(3). The DQGPM extends the classical power method framework into the algebra of dual quaternions—denoted $\hat{\mathbb{Q}}$—and provides convergence analysis, projection mechanisms, and algorithmic variants for both Hermitian and non-Hermitian dual quaternion matrices. Its foundation is the spectral theory of dual quaternion matrices, including the properties of unit dual quaternions, the behavior of their eigenvalues (which are dual numbers in the Hermitian case), and their applications to synchronization and rank-one matrix completion problems. The DQGPM achieves these tasks by iteratively applying matrix–vector products and normalizations adapted to dual quaternion algebra, with careful per-iteration projection onto constraint sets (e.g., unit dual quaternions) as required by the geometry of the problem (Cui et al., 2023, Chen et al., 21 May 2025, Yang et al., 1 Dec 2025, Chen et al., 2024, Zhao et al., 30 Jan 2026).

1. Mathematical Foundations: Dual Quaternions and Matrix Theory

A dual quaternion $\hat q \in \hat{\mathbb{Q}}$ is an element of the form $\hat q = \tilde{q}_{st} + \tilde{q}_I \epsilon$, where $\tilde{q}_{st}, \tilde{q}_I \in \mathbb{Q}$ (the skew-field of quaternions), $\epsilon^2 = 0$, and $\tilde{q}_{st}$ is called the standard part while $\tilde{q}_I$ is the infinitesimal (dual) part. The conjugate of $\hat q$ is $$\hat q^* = \tilde{q}_{st}^* + \tilde{q}_I^* \epsilon$$. The modulus is

$$|\hat q| = \begin{cases} |\tilde{q}_{st}| + \mathrm{sc}(\tilde{q}_{st}^* \tilde{q}_I) / |\tilde{q}_{st}|\,\epsilon, & \tilde{q}_{st} \neq 0, \ |\tilde{q}_I| \epsilon, & \tilde{q}_{st} = 0. \end{cases}$$

A unit dual quaternion satisfies $|\tilde{q}_{st}| = 1$ and $$\tilde{q}_{st} \tilde{q}_I^* + \tilde{q}_I \tilde{q}_{st}^* = 0$$, corresponding to rigid transformations in $\mathrm{SE}(3)$ (Cui et al., 2023, Zhao et al., 30 Jan 2026).

A dual quaternion matrix $\hat Q \in \hat{\mathbb{Q}}^{n \times n}$ is Hermitian if $\hat Q^* = \hat Q$. For Hermitian dual quaternion matrices, all right eigenvalues are dual numbers, and there exists a complete set of orthonormal dual quaternion eigenvectors (Cui et al., 2023, Chen et al., 2024, Chen et al., 21 May 2025).

2. Algorithmic Formulation of the DQGPM

For a Hermitian $\hat Q \in \hat{\mathbb{Q}}^{n \times n}$, the DQGPM seeks the dominant eigenpair $(\lambda_1, u_1)$ by the following iteration:

1. $\hat y^{(k)} = \hat Q \hat v^{(k-1)}$
2. $$\hat\lambda^{(k-1)} = (\hat v^{(k-1)})^* \hat y^{(k)}$$
3. Normalize: $$\hat v^{(k)} = P_\text{unit}(\hat y^{(k)}) = \hat y^{(k)}/\|\hat y^{(k)}\|_2$$, where normalization projects onto the unit sphere by analytic dual quaternion scaling.
4. Terminate if the residual $$\|\hat Q \hat v^{(k)} - \hat v^{(k)} \hat\lambda^{(k)}\|_{2^R}$$ is below a fixed tolerance.

In the non-Hermitian case, the iteration is syntactically similar, but the convergence theory is more delicate due to non-commutativity and potential eigenvalue non-existence. The algorithm's projection step is essential for enforcing problem-specific feasible sets, such as the unit dual quaternion group $\mathrm{UDQ}$ in synchronization tasks (Cui et al., 2023, Yang et al., 1 Dec 2025, Zhao et al., 30 Jan 2026).

3. Convergence Rates and Theoretical Guarantees

For Hermitian $\hat Q$, if the dominant eigenvalue $\lambda_1$ is strict (i.e., $|\lambda_{1,st}| > |\lambda_{2,st}|$), then DQGPM iterates converge to $(\lambda_1, u_1)$ linearly in the dual-number sense, with standard-part contraction rate $|\lambda_{2,st}/\lambda_{1,st}|$ and dual-part error decaying as the square (Cui et al., 2023, Chen et al., 21 May 2025): $$v^{(k)} = s^k\sum_{j=1}^l u_j \hat\gamma_j \left(1_D + \tilde O_D\left(\left|\frac{\lambda_{l+1,st}}{\lambda_{1,st}}\right|^k\right)\right), \quad s = \mathrm{sgn}(\lambda_{1,st}).$$ In the degenerate case when two or more eigenvalues share the same standard part but differ in the dual part, the standard DQGPM iteration cannot distinguish between them and may stall (Chen et al., 21 May 2025).

For non-Hermitian matrices, convergence requires additional algebraic-geometric spectral conditions. If the standard part has non-defective dominant eigenvalue and the appropriate dual-part spectral regularity holds, linear convergence can again be established, but failure of these conditions leads to lack of convergence or locking to incorrect invariant subspaces (Yang et al., 1 Dec 2025).

Robustness is inherited from classical power iteration: convergence is rapid if the spectral gap is large, and the method remains stable provided initialization has nontrivial projection onto the dominant eigenspace.

4. Variants: Dual-Complex Adjoint, Aitken Acceleration, and Full Spectrum Computation

Several algorithmic improvements and alternatives to raw DQGPM have appeared:

• Dual-Complex Adjoint Matrix Power Method (DCAM-PM): Maps the dual quaternion Hermitian matrix to a $2n \times 2n$ dual-complex Hermitian matrix, applies the complex power method, and recovers dual quaternion eigenvectors via a linear invertible transformation. DCAM-PM achieves comparable convergence rates but is more efficient in practice due to highly optimized complex arithmetic (Chen et al., 21 May 2025, Chen et al., 2024).
• Aitken-Accelerated DCAM-PM (ADCAM-PM): Applies Aitken extrapolation to the standard and dual parts of the iterates following DCAM-PM for super-linear convergence, reducing iteration count by 30–40\% (Chen et al., 21 May 2025).
• Eigen-Decomposition Dual-Complex Adjoint Method (EDDCAM-EA): Computes the entire spectrum of eigenpairs in $O(n^3)$ time by diagonalization of the dual-complex adjoint, then orthogonalizing and mapping back. EDDCAM-EA handles degenerate cases where DQGPM and DCAM-PM fail, including coinciding standard-part eigenvalues with distinct dual parts (Chen et al., 21 May 2025, Chen et al., 2024).

A table summarizing per-iteration costs:

Method	Per-iteration cost	Degenerate case handling
DQGPM	$O(n^2)$	No
DCAM-PM	$O(n^2)$	No
ADCAM-PM	$O(n^2)$ + $O(n)$	No, but faster in practice
EDDCAM-EA	$O(n^3)$	Yes; computes all eigenpairs

5. Projection Techniques and Feasibility Enforcement

Projection onto the set of unit dual quaternions plays a central role in DQGPM-based optimization. For any nonzero $\hat q = \tilde{q}_{st} + \tilde{q}_I \epsilon$,

$$P_{\mathrm{UDQ}}(\hat q) = \begin{cases} \frac{\hat q}{|\hat q|}, & \tilde{q}_{st} \ne 0, \ \frac{\tilde{q}_I}{|\tilde{q}_I|} + \hat u_I \epsilon, & \tilde{q}_{st} = 0,\,\tilde{q}_I \ne 0, \end{cases}$$

with explicit correction for the dual part. For vectors, projection is performed componentwise. The normalization is Lipschitz with constant 2, providing theoretical guarantees for error stability under perturbation (Cui et al., 2023, Zhao et al., 30 Jan 2026). In problems such as $\mathrm{SE}(3)$ synchronization, this projection step is embedded per-iteration to enforce feasibility with respect to the underlying Lie group structure and enables linear error contraction up to a noise-dependent floor.

6. Applications in SLAM, Synchronization, and Group Control

The DQGPM and its variants have been extensively applied to problems where spectral structure in the dual quaternion algebra is fundamental:

• Simultaneous Localization and Mapping (SLAM): Reformulated as a rank-one Hermitian dual quaternion completion, reduced to finding a dominant eigenpair subject to known entries and unit constraints. An alternating block-coordinate scheme couples closed-form Hermitian/unit-diagonal updates with best rank-one approximations computed by DQGPM (Cui et al., 2023).
• $\mathrm{SE}(3)$ Synchronization: The problem of registering absolute poses from noisy relative measurements is posed as maximizing $x^* C x$ over $x \in \mathrm{UDQ}^n$. The algorithm consists of a spectral (DQGPM-based) initializer, followed by DQGPM iterations with per-step unit dual quaternion projection. The method admits provable linear contraction of the standard-part error and achieves state-of-the-art accuracy and speed compared to EIG (spectral) and SDR (semidefinite) baselines. Residual errors decay to an explicit noise floor, and efficiency is demonstrated in large-scale synthetic and real 3D scan datasets (Zhao et al., 30 Jan 2026).
• Multi-agent Group Control: Laplacian matrices in formation control are often dual quaternion Hermitian, with DQGPM, DCAM-PM, and ADCAM-PM used to rapidly compute the dominant eigenpair, affecting convergence and stability properties of consensus algorithms (Chen et al., 21 May 2025, Chen et al., 2024).

7. Extensions, Limitations, and Practical Considerations

The DQGPM framework generalizes to non-Hermitian dual quaternion matrices, as in "The Power Method for Non-Hermitian Dual Quaternion Matrices" (Yang et al., 1 Dec 2025). Here, a Jordan-like decomposition for dual quaternion matrices is introduced, with necessary and sufficient spectral conditions for convergence. The method and analysis extend to dual complex and dual number matrices, the former via the dual-complex adjoint mapping.

Limiting factors include:

• Degenerate Eigenvalues: DQGPM fails to distinguish eigenvalues with identical standard parts and distinct dual components, requiring eigen-decomposition approaches.
• Spectral Gap Sensitivity: Like all power-type methods, the convergence rate depends critically on the standard-part spectral separation.
• Non-Hermitian Instabilities: In non-Hermitian cases, absence of the required algebraic-geometric spectral properties precludes convergence in general (Yang et al., 1 Dec 2025).

Empirical studies demonstrate that for well-conditioned instances (e.g., formation control, Laplacian matrices with large spectral gap, moderate noise in SLAM/SE(3)), DQGPM and its DCAM-PM/ADCAM-PM variants achieve rapid, robust convergence with low computational overhead (Cui et al., 2023, Chen et al., 21 May 2025, Chen et al., 2024, Zhao et al., 30 Jan 2026). In full-spectrum or degenerate regimes, EDDCAM-EA provides strong alternatives.

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Key References:

• "A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix" (Cui et al., 2023)
• "Improved power methods for computing eigenvalues of dual quaternion Hermitian matrices" (Chen et al., 21 May 2025)
• "The Power Method for Non-Hermitian Dual Quaternion Matrices" (Yang et al., 1 Dec 2025)
• "Applications of Dual Complex Adjoint Matrix in Eigenvalue Computation of Dual Quaternion Hermitian Matrix" (Chen et al., 2024)
• "Dual Quaternion SE(3) Synchronization with Recovery Guarantees" (Zhao et al., 30 Jan 2026)