🔢 mathematics

Iterative Methods for Computing the Moore--Penrose Inverse of Split-Quaternion Matrices with Applications

This paper establishes a consistent framework for computing the Moore-Penrose inverse of split-quaternion matrices via a real representation, analyzes the convergence of Newton-Schulz iterations with a novel polynomial initialization, and demonstrates their application to cross and CUR approximations.

Original authors: Salman Ahmadi-Asl, Valentin Leplat, Mohammad S. Alkousa

Published 2026-08-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Salman Ahmadi-Asl, Valentin Leplat, Mohammad S. Alkousa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a giant, multi-dimensional puzzle where the pieces aren't just flat squares, but tiny, four-sided spinning tops. In the world of mathematics, these tops are called "split-quaternions." You might know regular numbers, or even "complex" numbers that involve the square root of negative one. But split-quaternions are a wilder cousin. They are used to describe things like how light waves move or how objects twist in space-time, but they have a tricky secret: some of them are "zero divisors." In normal math, if you multiply two numbers and get zero, at least one of them must be zero. But with these special tops, you can multiply two non-zero pieces together and get a total nothingness. This makes doing the reverse operation—finding the "inverse" or the "undo" button—extremely difficult, especially when your puzzle pieces don't form a perfect square grid.

Scientists and engineers need to solve these puzzles to make better computer graphics, improve robot movements, and process signals from sensors. Usually, they turn these tricky 4D problems into simpler, flat 2D real-number problems to solve them. However, when the puzzle pieces are "broken" (rank-deficient) or contain those tricky zero divisors, the standard methods can get stuck or give the wrong answer. The big question is: how do we reliably find the "undo" button for these messy, broken, 4D puzzles without getting lost in the math?

This paper, titled "Iterative Methods for Computing the Moore–Penrose Inverse of Split-Quaternion Matrices with Applications," is like a master craftsman's guide to fixing those broken puzzles. The authors, Salman Ahmadi-Asl, Valentin Leplat, and Mohammad S. Alkousa, propose a new, step-by-step recipe to find the "Moore–Penrose inverse" (the perfect undo button) for these split-quaternion matrices.

First, they set up a clear rulebook. They show how to translate these 4D split-quaternion puzzles into a standard 2D real-number language using a specific "real representation." Think of this as translating a foreign language into English so you can use a standard dictionary. By doing this, they prove that even if the puzzle has broken pieces (zero divisors) or missing rows and columns, a unique "undo" button still exists and can be found.

The core of their discovery is a method called the "Newton–Schulz iteration." Imagine you are trying to guess the exact weight of a mystery box. You make a guess, check how far off you are, and then make a smarter guess based on that error. You repeat this, and with each step, your guess gets twice as accurate as the last. The authors show that this "guess-and-refine" method works perfectly for these split-quaternion puzzles, even when they are messy or incomplete. They proved mathematically that if you start with a reasonable guess, this method will always zoom in on the correct answer, no matter how broken the puzzle is.

To make this process even faster, they introduced a clever "warm-up" trick. Usually, you start with a simple, safe guess. But the authors found a way to use a low-degree polynomial (a fancy math curve) to make a much smarter initial guess. They call this the "LS–Gram polynomial initialization." It's like looking at the shape of the box before you even touch it to make a much better first guess. They tested this against other guessing strategies and found that while it takes a tiny bit of time to set up the curve, it saves a lot of time later by needing fewer steps to reach the perfect answer.

Finally, the authors showed how to use these new tools to build "CUR approximations." This is a way to summarize a huge, complex matrix by picking just a few key rows and columns and a small middle piece to reconstruct the whole thing. They proved that if you pick your rows and columns carefully, you can reconstruct the original puzzle exactly, or at least find the absolute best possible approximation.

In their computer experiments, the authors tested these methods on matrices of various sizes. They found that the new iterative method was significantly faster than the old, direct ways of solving the problem, often running three to four times faster in their tests. The "warm-up" polynomial trick helped reduce the number of steps needed to solve the puzzle, especially when the puzzles were very messy or "ill-conditioned" (very sensitive to small errors). The paper concludes that this new framework is a reliable, efficient, and mathematically sound way to handle these complex 4D matrices, opening the door for better applications in physics, robotics, and signal processing.

Technical Summary: Iterative Methods for Computing the Moore–Penrose Inverse of Split-Quaternion Matrices

Problem Statement
The paper addresses the computation of the Moore–Penrose (MP) inverse for matrices over the algebra of split-quaternions ( $\mathbb{Q}_s$ ). Unlike standard Hamilton quaternions, split-quaternions possess an indefinite quadratic form, leading to the existence of nontrivial zero divisors and nilpotent elements. These algebraic features complicate the definition of inverses and the application of standard spectral theory. Specifically, the indefinite nature of the quadratic form prevents the direct transfer of spectral results from real or complex matrices, and the presence of zero divisors necessitates careful handling of singular and rank-deficient cases. The authors aim to establish a robust framework for defining the MP inverse in this setting and developing efficient iterative algorithms for its computation, particularly for rectangular and rank-deficient matrices.

Methodology
The core methodological contribution is the establishment of a consistent framework based on a $2 \times 2$ real representation ( $\sigma$ ) and the $i$ -conjugate transpose ( $H$ ).

Real Representation: The authors define a bijective map $\sigma: \mathbb{Q}_s \to \mathbb{R}^{2 \times 2}$ that preserves multiplication and maps the $i$ -conjugate transpose to the standard real transpose. For a matrix $A \in \mathbb{Q}_s^{m \times n}$ , the real representative $A^\sigma \in \mathbb{R}^{2m \times 2n}$ allows the MP inverse to be defined directly via the real MP inverse: $A^\dagger = \sigma^{-1}((A^\sigma)^\dagger)$ . This definition clarifies the treatment of zero divisors and ensures the existence and uniqueness of the inverse.
Newton–Schulz Iteration: The paper analyzes the Newton–Schulz iteration for computing the MP inverse. Two implementations are presented:
- Embedded: Applying the standard real Newton–Schulz iteration to the real representative $A^\sigma$ .
- Native: A direct iteration in split-quaternion arithmetic ( $X_{k+1} = X_k(2I - AX_k)$ ).
  The authors prove that these two sequences are exactly equivalent under the representation $\sigma$ .
Convergence Analysis: Using a thin Singular Value Decomposition (SVD) of the real representative, the authors derive exact formulas for the iterates and residuals. They establish that the iteration converges to the MP inverse for matrices of arbitrary rank (including rank-deficient cases) provided the initial scaling parameter $\alpha$ satisfies $0 < \alpha < 2/\|A^\sigma\|_2^2$ . The analysis provides exact relations between the residual and the error, showing quadratic convergence of the residual.
Polynomial Initialization: To accelerate convergence, the authors propose a low-degree polynomial initialization inspired by Souriau's inverse recursion. This involves fitting a polynomial to the inverse of the normalized Gram factor ( $A^\sigma (A^\sigma)^T$ or $(A^\sigma)^T A^\sigma$ ) using a least-squares (LS) approach. The method includes a spectral acceptance test to ensure the initial residual lies within the convergence region, with a safe fallback to a scaled transpose if the test fails.

Key Contributions

Consistent Framework: The paper defines the MP inverse for split-quaternion matrices via the real representation and the $i$ -conjugate transpose, proving existence, uniqueness, and basic properties (including a scalar formula for zero divisors).
Convergence Theory: A complete convergence analysis for the Newton–Schulz iteration is provided for rectangular and rank-deficient matrices. The authors derive exact error recurrences and residual-error identities, removing the full-rank restrictions found in previous analyses of Hamilton quaternion matrices.
Equivalence of Implementations: The paper demonstrates the exact equivalence between the embedded (real) and native (split-quaternion) Newton–Schulz iterations, validating the use of native arithmetic without loss of theoretical guarantees.
LS–Gram Initialization: A novel polynomial initialization is introduced that adapts to the specific spectrum of the Gram matrix. It is shown to reduce the number of iterations required compared to standard scaled-transpose initialization.
CUR Approximation: The authors extend cross and CUR approximation theory to split-quaternion matrices. They prove that $C^\dagger A R^\dagger$ is the Frobenius-optimal middle factor for fixed sampled rows and columns. Furthermore, they establish a rank-revealing condition under which the cheaper cross factor $W^\dagger$ (where $W$ is the intersection matrix) yields an exact reconstruction.

Results
Numerical experiments conducted in MATLAB and Python validate the theoretical findings:

Accuracy: The iterative methods achieve accuracy comparable to direct SVD-based computation (relative errors around $10^{-11}$ to $10^{-14}$ ).
Efficiency: For medium-sized dense matrices, the Newton–Schulz iteration is significantly faster than direct SVD computation (e.g., ~3.7x faster for a $200 \times 250$ matrix).
Initialization Performance: The LS–Gram polynomial initialization consistently reduces the number of Newton–Schulz iterations compared to a scaled transpose. In experiments with condition numbers up to $10^4$ , the LS–Gram method reduced iterations by approximately 20% relative to the scaled transpose and remained competitive with a Chebyshev polynomial initialization that required exact spectral bounds.
CUR Approximation: The experiments confirm that the cross approximation using $W^\dagger$ can achieve exact reconstruction for low-rank matrices when the rank-revealing condition is met, with the iterative computation of the core inverse being substantially faster than direct methods.

Significance and Claims
The paper claims to provide the first consistent framework for the MP inverse of split-quaternion matrices that explicitly handles zero divisors and rank deficiency through a real representation. By transferring the analysis of Newton–Schulz iterations from real matrices to the split-quaternion setting, the authors establish rigorous convergence guarantees for arbitrary rank matrices. The proposed LS–Gram initialization offers a practical improvement over standard initialization strategies, reducing computational cost without requiring prior knowledge of the spectral bounds (unlike Chebyshev methods). Finally, the extension of CUR approximation results provides a theoretical foundation for low-rank approximations in split-quaternion signal processing and physics applications, characterizing the optimality of specific factors and conditions for exact reconstruction. The authors note that while the methods are efficient, the primary contribution is the theoretical unification and the development of robust iterative solvers, with future work suggested for sparse matrices and the Drazin inverse.

More like this