Computing Left Eigenvalues of Quaternion Matrices

Abstract: We present a practical Newton-based method for computing left eigenvalues of quaternion matrices. It uses only standard real/complex linear-algebra kernels via embeddings and applies to matrices of any size. Extensive tests on literature examples and benchmark ensembles, together with a compact MATLAB reference implementation, demonstrate reproducible, certificate-based computations up to size 64x64, including the detection of multiple spherical components and non-generic phenomena such as more than n isolated left eigenvalues and left-spectrum deficiency.

• The paper introduces a novel Newton-based algorithm for computing left eigenvalues of quaternion matrices, addressing the challenges of left spectrum continuity and minimizing the risk of missing eigenvalues.
• These left eigenvalues are computed via a gauged Newton iteration, fixing the right-scaling orbit while solving a well-conditioned linear system, ensuring local quadratic convergence under simplicity conditions.
• The method validates on a variety of matrix types, showing full recovery of isolated eigenvalues for both generic and non-generic cases, with a benchmark covering sizes from 2x2 to 64x64.
• follow_up_questions
• Can the Newton-based algorithm be extended to compute left eigenvalues of even larger quaternion matrices?
• To what extent can the methods demonstrated be adapted for other non-commutative algebras like octonions?
• How do the gauge constraints influence the performance of Newton iterations for matrices with multiple or defect eigenvectors?
• Find recent papers about quaternion matrix spectral theory.
• The method allows determination of specifics of the spectrum, how is this method related to techniques that don’t invoke principal components.
• Can the method be extended to compute right eigenvalues of other non-commutative matrix types?

Overview

The paper develops a practical Newton-based algorithm for computing left eigenvalues of quaternion matrices, i.e., quaternions $\lambda$ for which $Ax = \lambda x$ admits a nonzero solution. Left eigenvalues are substantially harder to compute than right eigenvalues: the left spectrum is not invariant under quaternionic similarity, it may contain continuous components (2-spheres), and its cardinality can deviate from the generic count of $n$ isolated points in either direction. The author's approach applies Newton's method directly to the eigenpair defect $Ax - \lambda x = 0$, augmented by a gauge constraint that fixes the four real degrees of freedom of eigenvector right-scaling, and reduces each correction step to a single square real linear system via a real embedding $\rho(\cdot)$. A compact MATLAB reference implementation is released, and benchmarks demonstrate certified computations up to size $64 \times 64$.

Background and problem structure

The paper first assembles the necessary spectral theory. Every quaternion matrix has at least one left eigenvalue (Wood's theorem), the left spectrum $\sigma_\ell(A)$ is compact in $\mathbb{H} \simeq \mathbb{R}^4$, and for a fixed nonzero eigenvector the left eigenvalue is unique, while eigenvectors are defined only up to right scaling $x \mapsto xq$. For $2\times 2$ matrices, Huang and So established a complete classification: one, two, or infinitely many (spherical) left eigenvalues. Beyond dimension three, no complete classification exists — a gap the numerical evidence in this paper illustrates rather than closes.

A key structural observation motivates avoiding determinantal elimination: since $Ax = \lambda x$0, the spectral parameter enters blockwise through the $Ax = \lambda x$1 left-multiplication matrix $Ax = \lambda x$2, so eliminating the eigenvector yields a single real polynomial in four variables of degree up to $Ax = \lambda x$3. Expanding or globally solving this system becomes impractical with growing $Ax = \lambda x$4. The paper instead keeps the eigenvector as an unknown and replaces global elimination by a sequence of well-conditioned linear solves inside a local iteration.

The gauged Newton iteration

The core technical device is a gauge constraint fixing the right-scaling orbit. With pivot index $Ax = \lambda x$5, the solver imposes $Ax = \lambda x$6 and $Ax = \lambda x$7; an explicit regauging lemma shows every eigenvector with nonzero $Ax = \lambda x$8-th entry can be brought into this canonical form without changing $Ax = \lambda x$9. The Newton correction solves

$n$0

a square $n$1 real system, where $n$2 encodes left multiplication $n$3 and $n$4 encodes the linearized gauge constraints. The author proves local quadratic convergence under a natural simplicity condition — geometric multiplicity one plus $n$5 — via a clean injectivity argument on the homogeneous Jacobian equations. Notably, the proof shows the gauge constraints eliminate exactly the residual right-scaling freedom, which is what makes the Jacobian invertible.

For spherical components, the situation is qualitatively different: the paper proves that any non-isolated zero of the gauged system has singular Jacobian, and gives an explicit $n$6 example where a nonzero homogeneous Jacobian solution exists. No quadratic convergence or uniqueness claim is made there; instead, multi-start runs are used to sample points on the continuum, followed by clustering and robust sphere fitting in $n$7 with outlier rejection and residual-based certification. This yields reproducible evidence for spherical structure but carries no completeness guarantee — Newton cannot enumerate a continuum.

Certification and refinement

Two scalar certificates are used throughout: the eigenpair residual $n$8 and the vector-free minimal residual $n$9, both reported absolutely and normalized by $Ax - \lambda x = 0$0. A two-stage polishing step — derivative-free local minimization of $Ax - \lambda x = 0$1 over $Ax - \lambda x = 0$2, then damped Newton correction on the eigenpair — routinely drives residuals to near machine precision. This certificate-based reporting is what allows the paper's claims about degenerate cases to be checked independently.

Validation

The method reproduces all closed-form literature spectra accurately. On the Huang–So examples, computed eigenvalues match $Ax - \lambda x = 0$3 and the explicit non-real pair up to roundoff, with normalized minimal-residual certificates at the $Ax - \lambda x = 0$4–$Ax - \lambda x = 0$5 level; on the spherical example, sphere fitting from 20 samples recovers center 2 and radius 1, with post-refinement inlier certificates at $Ax - \lambda x = 0$6–$Ax - \lambda x = 0$7. On the Macías-Virgós–Pereira-Sáez $Ax - \lambda x = 0$8 examples, the solver confirms known spectra, verifies that the pole $Ax - \lambda x = 0$9 is not a left eigenvalue of one matrix, reproduces rank-3 and maximal-rank differential diagnostics, and matches the null-rank degeneracy of a left-spectrum-deficient case — where reduced componentwise accuracy ($\rho(\cdot)$0 deviations) is consistent with the extreme degeneracy rather than a solver failure.

The benchmark suite covers dense i.i.d., Hermitian, triangular, and sparse families from $\rho(\cdot)$1 to $\rho(\cdot)$2, requesting $\rho(\cdot)$3 distinct eigenpairs per matrix. Across all sizes and families, full recovery ($\rho(\cdot)$4, acceptance rate 100%) was achieved on essentially all sampled instances. Sparse inputs are the most variable, requiring careful restart budgeting. Two caveats apply: the benchmark targets the generic case of $\rho(\cdot)$5 isolated eigenvalues and does not systematically trigger continua or deficiency, and success is defined relative to a prescribed trial budget rather than proven globally.

Non-generic phenomena

The most striking results concern spectra violating the generic picture. A $\rho(\cdot)$6 integer quaternion matrix is exhibited with five distinct isolated left eigenvalues — more than $\rho(\cdot)$7 — certified with $\rho(\cdot)$8 and minimum pairwise separation near 3.0. Conversely, a $\rho(\cdot)$9 construction converges consistently to only two isolated solutions ($64 \times 64$0), demonstrating left-spectrum deficiency; the paper notes that distinctness counting here is sensitive to the de-duplication tolerance, with clouds around each representative requiring a tolerance around $64 \times 64$1 to collapse correctly. Finally, a constructed $64 \times 64$2 matrix exhibits a mixed configuration: a detected spherical component with center $64 \times 64$3 and radius 8 (18 inliers, median normalized certificate $64 \times 64$4) coexisting with two isolated eigenvalues. These examples indicate that the low-dimensional classification does not extend to higher dimensions, though they constitute numerical evidence rather than proofs of the underlying spectral structure.

Limitations and open questions

The paper is candid about scope. Convergence theory covers only simple, isolated, gauged eigenpairs; behavior at multiple or defective eigenvalues is empirical only, as illustrated by the reduced accuracy at the null-rank eigenvalue. Sphere detection provides samples, not completeness — the number of spherical components or their exact parameterization cannot be certified from finite sampling. The benchmark protocol presupposes exactly $64 \times 64$5 isolated eigenvalues, so failure modes under heavy degeneracy are characterized only through the curated "Zoo" examples. The de-duplication threshold is a user-chosen parameter whose effect on cardinality counts is acknowledged but not resolved. Open questions include sharp upper bounds on the number of isolated left eigenvalues for general $64 \times 64$6, a classification of mixed spherical-plus-isolated configurations, and a posteriori error bounds linking residual certificates to eigenvalue perturbation under ill-conditioning.

Conclusion

The paper delivers a working computational pipeline for quaternionic left eigenvalues: a gauge-fixed Newton core with provable local quadratic convergence at simple isolated eigenpairs, embedded entirely in standard real linear algebra, wrapped in multi-start acceptance/de-duplication logic with residual certificates and optional polishing. Its principal value is twofold — reliable computation at moderate scale (up to $64 \times 64$7 with machine-precision certificates), and the first systematic numerical access to non-generic left-spectral phenomena such as more than $64 \times 64$8 isolated eigenvalues, spectrum deficiency, and mixed spherical configurations, which currently lack theoretical classification.