Quaternion Eigenproblem Analysis

• Quaternion-Based Eigenproblem is the spectral analysis of matrices with quaternion entries, featuring noncommutative arithmetic and distinct left/right eigenvalue definitions.
• Real embeddings, permutation-based determinants, and quasi-determinant frameworks reduce quaternion eigenproblems to tractable real or complex systems for efficient computation.
• Advanced algorithms like Newton-based iterations and structure-preserving QR methods achieve high precision, scaling for applications in optimization, robotics, and pose estimation.

A quaternion-based eigenproblem concerns the spectral analysis of matrices with entries in the noncommutative division ring of real quaternions. Central to this field are both left and right eigenproblems, each exhibiting algebraic, computational, and geometric phenomena not present in classical real or complex linear algebra. This article provides a detailed and technically rigorous synthesis of methodologies and results for spectral theory and computation with quaternion and dual-quaternion matrices, including left/right eigenvalue definitions, determinant concepts, canonical forms, algorithms, and applications to optimization and engineering.

1. Algebraic Structure: Left and Right Eigenvalue Problems

A quaternion $q \in \mathbb{H}$ is $q = q_0 + q_1 i + q_2 j + q_3 k$, with $q_0, ..., q_3 \in \mathbb{R}$ and $i^2 = j^2 = k^2 = ijk = -1$. Quaternionic matrices inherit the noncommutativity of $\mathbb{H}$. For $A \in \mathbb{H}^{n \times n}$:

• Left eigenvalue problem: Find $\lambda \in \mathbb{H}$ and $x \neq 0$ such that $A x = \lambda x$.
• Right eigenvalue problem: Find $\mu \in \mathbb{H}$ and $q = q_0 + q_1 i + q_2 j + q_3 k$0 such that $q = q_0 + q_1 i + q_2 j + q_3 k$1.

Left and right eigenvalues are generally disjoint; right eigenvalues are determined up to similarity classes: $q = q_0 + q_1 i + q_2 j + q_3 k$2, while left eigenvalues are direct quaternionic values, but only real-similarity $q = q_0 + q_1 i + q_2 j + q_3 k$3 with $q = q_0 + q_1 i + q_2 j + q_3 k$4 preserves the left spectrum, not full quaternionic similarity (Kou et al., 2015, Kou et al., 2016, Sebek, 2 Feb 2026).

In dual quaternionic settings (for applications such as spatial rigid-body kinematics), matrix entries take values in $q = q_0 + q_1 i + q_2 j + q_3 k$5, with $q = q_0 + q_1 i + q_2 j + q_3 k$6 and dual number parts. Dual quaternion Hermitian matrices have eigenspectra comprising dual numbers (Qi et al., 2021).

2. Determinants and Canonical Matrix Representations

Noncommutativity requires nonstandard notions of determinants:

• Permutation-based determinant $q = q_0 + q_1 i + q_2 j + q_3 k$7: Employs cycle decompositions for unambiguous definition, essential for right eigenvalue characterization (Kou et al., 2016).
• Real and complex embeddings: Maps quaternionic matrices into real or complex block-matrices (e.g., $q = q_0 + q_1 i + q_2 j + q_3 k$8 or complex adjoint), maintaining structure to exploit standard linear algebra kernels (Liu et al., 2019, Jia et al., 2017, Shiozaki, 2015).
• Quasi-determinant (dual/quaternion case): Embeds dual quaternion matrices into dual-complex block structure and defines $q = q_0 + q_1 i + q_2 j + q_3 k$9 with $q_0, ..., q_3 \in \mathbb{R}$0, crucial for spectral theory of dual quaternion matrices (Ling et al., 2024).

These constructions permit reduction of quaternion/dual quaternion eigenproblems to the solution of real or complex polynomial systems or block linear algebra problems, while preserving the algebraic features necessary for correct spectral computation.

3. Solution Methods for Quaternion-Based Eigenproblems

3.1 Left Eigenvalue Computation

Classical polynomial methods fail because $q_0, ..., q_3 \in \mathbb{R}$1 is noncommutative and there is no valid characteristic polynomial. Instead, the left eigenvalue problem for $q_0, ..., q_3 \in \mathbb{R}$2 is recast via a $q_0, ..., q_3 \in \mathbb{R}$3 real representation $q_0, ..., q_3 \in \mathbb{R}$4 (one of 48 possible “Hamiltonian” bases), leading to a block $q_0, ..., q_3 \in \mathbb{R}$5 real matrix $q_0, ..., q_3 \in \mathbb{R}$6. The condition for $q_0, ..., q_3 \in \mathbb{R}$7 being a left eigenvalue is equivalent to the singularity of $q_0, ..., q_3 \in \mathbb{R}$8.

The vanishing of four specific $q_0, ..., q_3 \in \mathbb{R}$9 minors $i^2 = j^2 = k^2 = ijk = -1$0, each a real polynomial of degree $i^2 = j^2 = k^2 = ijk = -1$1 in the four real components of $i^2 = j^2 = k^2 = ijk = -1$2, yields a well-determined algebraic system for all left eigenvalues (Liu et al., 2019). Numerical solution uses Groebner, resultants, or Newton-based solvers in $i^2 = j^2 = k^2 = ijk = -1$3.

3.2 Newton-Based Left-Eigenvalue Algorithm

For generic $i^2 = j^2 = k^2 = ijk = -1$4, a recent Newton framework solves the augmented system $i^2 = j^2 = k^2 = ijk = -1$5, $i^2 = j^2 = k^2 = ijk = -1$6 (gauge-fixing to mod out right-scaling) in real coordinates via embeddings, enabling local quadratic convergence for simple isolated eigenvalues. The multi-start strategy recovers both isolated and continuous (e.g., spherical) spectral components, with residual-based certificates validating output (Sebek, 2 Feb 2026).

3.3 Right Eigenvalue Computation

For right eigenvalues, the standard approach is to reduce $i^2 = j^2 = k^2 = ijk = -1$7 to quaternionic upper triangular (Schur-type) form using structure-preserving unitaries. Diagonal entries then yield representatives for all right similarity classes. For Hermitian matrices, the right eigenvalues are real, and an orthonormal right-eigenvector basis exists (Kou et al., 2015).

For general dual-quaternion Hermitian matrices, right eigenvalues are always dual numbers. The right eigenvalues are computed as roots of the quasi-characteristic polynomial via dual-complex block embeddings (Ling et al., 2024), or by mapping to dual-complex adjoint matrices and solving standard spectral problems in $i^2 = j^2 = k^2 = ijk = -1$8 (Chen et al., 2024, Chen et al., 2024).

4. Numerical Algorithms and Structure-Preserving Decompositions

4.1 Block Real and Dual-Complex Approaches

• JRS-QR and Real Counterpart Embeddings: The JRS symmetry framework enables an entirely real-arithmetic approach for the right eigenproblem of general quaternion matrices, with an embedding that preserves unitary and Hermitian/JRS-symplectic structure. Structure-preserving QR iterations (Francis double-shift, Householder, Givens) yield backward stable and efficient Schur form computations with superior performance over naïve embeddings (Jia et al., 2017).
• Blocked Paige–Van Loan Algorithm: For very large Hermitian quaternionic matrices (as arise in relativistic quantum chemistry), the Paige–Van Loan blocked tridiagonalization and subsequent diagonalization, exploiting block structure and Level 3 BLAS, enables practical solution for dimensions $i^2 = j^2 = k^2 = ijk = -1$9 (Shiozaki, 2015).
• Non-Hermitian Cases: Quaternionic Hessenberg-to-Schur (QR) algorithms with eigenvector extraction and eigenvalue reordering techniques (aggressive early deflation, AED) extend high-performance computation to general non-Hermitian matrices (Jia et al., 4 Nov 2025).

4.2 Dual Quaternion Hermitian Methods

Spectral theory for dual quaternion Hermitian matrices is closely analogous to the real and quaternionic cases but with dual numbers as eigenvalues. Unitary diagonalization holds, and algorithms such as the dual quaternion power method, Rayleigh quotient iteration (RQI), and the generalized Jacobi method are available:

• Power Method and RQI: The power method has linear convergence when the dominance condition on standard parts is met (Cui et al., 2023). RQI achieves local cubic convergence and minimal-residual properties for simple eigenvalues (Duan et al., 2023, Chen et al., 2024).
• Jacobi-Type Algorithms: In cases with eigenvalues equal in standard part but distinguished in dual part (when standard iterative methods fail), the three-step generalized Jacobi method is finite, robust, and achieves $\mathbb{H}$0-accuracy in both standard and dual components (Chen et al., 2024).
• Dual-Complex Adjoint Matrix Methods: The dual-complex adjoint transforms dual quaternion Hermitian eigenvalue problems to dual-complex spectral decompositions. This enables both single and full spectrum computation with improved arithmetic efficiency and outperforms direct dual-quaternion arithmetic (Chen et al., 2024).

5. Applications: Alignment, Toeplitz Matrices, and Robotics

5.1 Rotation, Pose Alignment, and Adjugate Methods

Quaternion-based eigensystems are fundamental in geometric optimization problems such as the orthogonal Procrustes problem for 3D rotations, pose extraction, and frame alignment in robotics and computer vision. The optimal rotation minimizing RMSD between point clouds is given by the eigenvector corresponding to the maximal eigenvalue of a symmetric $\mathbb{H}$1 profile matrix constructed from the data covariance (Hanson, 2018). Closed-form algebraic solutions (quartic eigenpolynomial, Cardano/Ferrari method) are available, as are adjugate-matrix-based algorithms for extracting quaternions from rotation matrices, which manage the multi-valued nature of quaternion representations and avoid singularities by covering the quaternionic sphere with adjugate patches (Hanson et al., 2022).

5.2 Toeplitz Structure and Fast Linear System Solution

Hermitian quaternion Toeplitz matrices generated by quaternion-valued functions decompose via block-complex representations, allowing analysis of their spectrum via a quaternionic Grenander–Szegö theorem (Lin et al., 21 Apr 2025). Efficient diagonalization of Hermitian quaternion circulants is achieved with quaternionic DFT matrices. Applications include fast preconditioned conjugate gradient methods (PCG) with $\mathbb{H}$2 arithmetic complexity per iteration for large signal-processing problems.

5.3 Robotics, Pose Graph Optimization, and Control

Dual quaternion matrices are natural for encoding rigid-body pose and kinematic relationships. Spectral algorithms for dual quaternion Hermitian matrices underpin coordinate alignment, pose estimation, and pose graph optimization, with efficient eigendecomposition essential for SLAM and formation control applications (Qi et al., 2021, Cui et al., 2023, Chen et al., 2024). The introduction of dual-complex adjoint methods has yielded significant improvements in speed and accuracy for these large-scale, high-dimensional problems.

6. Fundamental Theoretical Results and Spectral Properties

• Existence: Every square quaternionic matrix has at least one left eigenvalue (Wood's theorem) (Liu et al., 2019, Sebek, 2 Feb 2026).
• Multiplicity and Continuous Spectrum: The left spectrum may include isolated points or higher-dimensional manifolds (most commonly spheres) and may contain more or fewer than $\mathbb{H}$3 points. Newton-based solvers can detect both discrete and spherical spectral phenomena (Sebek, 2 Feb 2026, Liu et al., 2019).
• Spectral Inclusion Theorems: Bounds and Gershgorin-type inclusion sets (in the quaternionic case, 3D balls) apply to left eigenvalues (Liu et al., 2019).
• Similarity Invariance: Right eigenvalues are similarity classes; left eigenvalues are invariant under real similarity only (Kou et al., 2015, Sebek, 2 Feb 2026).
• Spectral Theorems for Dual Quaternion Hermitians: There exists a unitary basis and ordered dual-number eigenvalues; matrix positive (semi)definiteness is determined by the total order on dual numbers (Qi et al., 2021).
• Determinantal Identities: For dual quaternion Hermitian matrices, the quasi-determinant equals the product of squared magnitudes of all eigenvalues (Ling et al., 2024), paralleling classical Hermitian theory.

7. Numerical Precision, Complexity, and Software

Numerical algorithms for quaternion-based eigenproblems are capable of high precision (residuals down to $\mathbb{H}$4), cubic convergence (RQI), and efficient scaling to $\mathbb{H}$5 in optimized implementations. Structure-aware methods (real and complex embeddings, block structures, JRS-symmetry) yield order-of-magnitude speedups over naïve algorithms (Jia et al., 2017, Shiozaki, 2015, Chen et al., 2024, Chen et al., 2024). Reference implementations and open source code for both classical and modern algorithms exist in multiple languages and frameworks.

---

Table: Core Methods for Quaternion-Based Eigenproblems

Matrix Type	Eigenproblem Type	Core Methods
General Quaternion	Left	Real embedding, 4-poly system, Newton-based multi-start (Liu et al., 2019, Sebek, 2 Feb 2026)
General Quaternion	Right	Schur form, triangularization, permutation determinant (Kou et al., 2015, Kou et al., 2016, Jia et al., 2017)
Hermitian Quaternion	Right	Real eigenproblem, adjugate/identity-based eigenvector extraction (He et al., 2022, Kou et al., 2015)
Dual Quaternion Hermitian	Right	Quasi-determinant, dual-complex adjoint, RQI, Jacobi (Qi et al., 2021, Ling et al., 2024, Duan et al., 2023, Chen et al., 2024, Chen et al., 2024, Chen et al., 2024)

---

• (Liu et al., 2019) Quaternionic left eigenvalue problem: a matrix representation
• (Sebek, 2 Feb 2026) Computing Left Eigenvalues of Quaternion Matrices
• (Kou et al., 2015) Linear Quaternion Differential Equations: Basic Theory and Fundamental Results
• (Kou et al., 2016) Solve the linear quaternion-valued differential equations having multiple eigenvalues
• (Jia et al., 2017) A New Real Structure-preserving Quaternion QR Algorithm
• (Shiozaki, 2015) An efficient solver for large structured eigenvalue problems in relativistic quantum chemistry
• (Jia et al., 4 Nov 2025) On Eigenvector Computation and Eigenvalue Reordering for the Non-Hermitian Quaternion Eigenvalue Problem
• (Hanson, 2018) The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems
• (Hanson et al., 2022) Exploring the Adjugate Matrix Approach to Quaternion Pose Extraction
• (Lin et al., 21 Apr 2025) Hermitian Quaternion Toeplitz Matrices by Quaternion-valued Generating Functions
• (Qi et al., 2021) Eigenvalues and Singular Values of Dual Quaternion Matrices
• (Ling et al., 2024) Quasi-determinant and right eigenvalues of dual quaternion matrices
• (Duan et al., 2023) On Rayleigh Quotient Iteration for Dual Quaternion Hermitian Eigenvalue Problem
• (Cui et al., 2023) A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix
• (Chen et al., 2024) Generalized Jacobi Method for Computing Eigenvalues of Dual Quaternion Hermitian Matrices
• (Chen et al., 2024) Dual Complex Adjoint Matrix: Applications in Dual Quaternion Research
• (Chen et al., 2024) Applications of Dual Complex Adjoint Matrix in Eigenvalue Computation of Dual Quaternion Hermitian Matrix
• (He et al., 2022) The eigenvector-eigenvalue identity for the quaternion matrix with its algorithm and computer program