Quaternion Matrix Research Articles

New families of quaternionic Hadamard matrices

A quaternionic Hadamard matrix (QHM) of order n is an n×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\ imes n$$\\end{document} matrix H with non-zero entries in the quaternions such that HH∗=nIn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$HH^*=nI_n$$\\end{document}, where In\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_n$$\\end{document} and H∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^*$$\\end{document} denote the identity matrix and the conjugate-transpose of H, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements.

A new structure-preserving algorithm based on the semi-tensor product of matrices for split quaternion matrix LDU decomposition and its applications

In this paper, we put forward a new structure-preserving algorithm based on the semi-tensor product of matrices for the split quaternion matrix LDU decomposition, which makes full use of high-level operations, and relation of operations between split quaternion matrices and their L-representation matrices. The algorithm is more stable and efficient than the complex structure-preserving algorithm proposed in Wang et al. (2021). Numerical experiments are provided to verify the efficiency of the new structure-preserving algorithm.

Schur analysis over the unit spectral ball

We begin a study of Schur analysis when the variable is now a matrix rather than a complex number. We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of hypercomplex numbers.

A Dynamic Parameter Noise-Tolerant Zeroing Neural Network for Time-Varying Quaternion Matrix Equation With Applications.

As a common and significant problem in the field of industrial information, the time-varying quaternion matrix equation (TV-QME) is considered in this article and addressed by an improved zeroing neural network (ZNN) method based on the real representation of the quaternion. In the light of an improved dynamic parameter (IDP) and an innovative activation function (IAF), a dynamic parameter noise-tolerant ZNN (DPNTZNN) model is put forward for solving the TV-QME. The presented IDP with the character of changing with the residual error and the proposed IAF with the remarkable performance can strongly enhance the convergence and robustness of the DPNTZNN model. Therefore, the DPNTZNN model possesses fast predefined-time convergence and superior robustness under different noise environments, which are theoretically analyzed in detail. Besides, the provided simulative experiments verify the advantages of the DPNTZNN model for solving the TV-QME, especially compared with other ZNN models. Finally, the DPNTZNN model is applied to image restoration, which further illustrates the practicality of the DPNTZNN model.

Solution to Several Split Quaternion Matrix Equations

Split quaternions have various applications in mathematics, computer graphics, robotics, physics, and so on. In this paper, two useful, real representations of a split quaternion matrix are proposed. Based on this, we derive their fundamental properties. Then, via the real representation method, we obtain the necessary and sufficient conditions for the existence of solutions to two split quaternion matrix equations. In addition, two experimental examples are provided to show their feasibility.

A new Sylvester-type quaternion matrix equation model for color image data transmission

A new Sylvester-type quaternion matrix equation model for color image data transmission

Fast Eigenvalue Decomposition of Arrowhead and Diagonal-Plus-Rank-k Matrices of Quaternions

Quaternions are a non-commutative associative number system that extends complex numbers, first described by Hamilton in 1843. We present algorithms for solving the eigenvalue problem for arrowhead and DPRk (diagonal-plus-rank-k) matrices of quaternions. The algorithms use the Rayleigh Quotient Iteration with double shifts (RQIds), Wielandt’s deflation technique and the fact that each eigenvector can be computed in O(n) operations. The algorithms require O(n2) floating-point operations, n being the order of the matrix. The algorithms are backward stable in the standard sense and compare well to the standard QR method in terms of speed and accuracy. The algorithms are elegantly implemented in Julia, using its polymorphism feature.

Investigation of some Sylvester-type quaternion matrix equations with multiple unknowns

Investigation of some Sylvester-type quaternion matrix equations with multiple unknowns

Quaternion matrix completion using untrained quaternion convolutional neural network for color image inpainting

The use of quaternions as a novel tool for color image representation has yielded impressive results in color image processing. By considering the color image as a unified entity rather than separate color space components, quaternions can effectively exploit the strong correlation among the RGB channels, leading to enhanced performance. Especially, color image inpainting tasks are highly beneficial from the application of quaternion matrix completion techniques, in recent years. However, existing quaternion matrix completion methods suffer from two major drawbacks. First, it can be difficult to choose a regularizer that captures the common characteristics of natural images, and sometimes the regularizer that is chosen based on empirical evidence may not be the optimal or efficient option. Second, the optimization process of quaternion matrix completion models is quite challenging because of the non-commutativity of quaternion multiplication. To address the two drawbacks of the existing quaternion matrix completion approaches mentioned above, this paper tends to use an untrained quaternion convolutional neural network (QCNN) to directly generate the completed quaternion matrix. This approach replaces the explicit regularization term in the quaternion matrix completion model with an implicit prior that is learned by the QCNN. Extensive quantitative and qualitative evaluations demonstrate the superiority of the proposed method for color image inpainting compared with some existing quaternion-based and tensor-based methods.

A novel structure-preserving algorithm for the singular value decomposition of biquaternion matrices

In this paper, we study the singular value decomposition of biquaternion matrices. We prove that the singular value decomposition of biquaternion matrices can be equivalently converted to the singular value decomposition of quaternion adjoint matrices. By means of the Cheng product, we propose a novel representation called Q -representation and design a Q structure-preserving algorithm to deal with the singular value decomposition of quaternion matrices, which makes use of high-level operations. The algorithm is more efficient than that in quaternion toolbox for matlab using quaternion arithmetics and real structure-preserving algorithm. The Q structure-preserving algorithm for singular value decomposition of biquaternion matrices is presented through the application of the Q structure-preserving algorithm for singular value decomposition of quaternion matrices. Numerical experiments are provided to demonstrate the efficiency of the Q structure-preserving algorithm. We apply the Q structure-preserving algorithm of quaternion singular value decomposition to improve the image denoising process.

Fejér–Riesz factorization in the QRC-subalgebra and circularity of the quaternionic numerical range

We provide a characterization when the quaternionic numerical range of a matrix is a closed ball with center 0. The proof makes use of Fejér–Riesz factorization of matrix-valued trigonometric polynomials within the algebra of complex matrices associated with quaternion matrices.

Universality of spectral fluctuations in open quantum chaotic systems

Quantum chaotic systems with one-dimensional spectra follow spectral correlations of Orthogonal (OE), Unitary (UE), or Symplectic Ensembles (SE) of random matrices depending on their invariance under time reversal and rotation. In this letter, we study the non-Hermitian and non-unitary ensembles based on the symmetry of matrix elements, viz. ensemble of complex symmetric, complex asymmetric (Ginibre), and self-dual matrices of complex quaternions. The eigenvalues for these ensembles lie in the two-dimensional plane. We show that the fluctuation statistics of these ensembles are universal and quantum chaotic systems belonging to OE, UE, and SE in the presence of a dissipative environment show similar spectral fluctuations. The short-range correlations are studied using spacing ratio and spacing distribution. For long-range correlations, unfolding at a non-local scale is crucial. We describe a generic method to unfold the two-dimensional spectra with non-uniform density and evaluate correlations using number variance. We find that both short-range and long-range correlations are universal. We verify our results with the quantum kicked top in a dissipative environment that can be tuned to exhibit symmetries of OE, UE, and SE in its conservative limit.

The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples.

Fixed-precision randomized quaternion singular value decomposition algorithm for low-rank quaternion matrix approximations

The fixed-precision randomized quaternion singular value decomposition algorithm (FPRQSVD) is presented to compute the low-rank quaternion matrix approximation. The FPRQSVD algorithm estimates the appropriate rank according to the given tolerance without a predetermined rank parameter, and computes the corresponding low-rank quaternion matrix approximation automatically. Error analysis indicates that the FPRQSVD algorithm is mathematically equivalent to the randomized quaternion singular value decomposition algorithm. Numerical experiments are given to demonstrate that the FPRQSVD algorithm is feasible and effective. Moreover, we use it to deal with the problem of color image inpainting.

Dual Quaternion Matrix Equation AXB = C with Applications

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation AXB=C has been extensively studied. However, there is currently limited information on matrix equation AXB=C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation AXB=C, and present the expression for the general solution when it is solvable. As an application, we derive the ϕ-Hermitian solutions for dual quaternion matrix equation AXAϕ=C, where the ϕ-Hermitian extends the concepts of Hermiticity and η-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.

CUR and Generalized CUR Decompositions of Quaternion Matrices and their Applications

Low-rank approximations of quaternion matrices have garnered interest across various applications, such as color images and signal processing. In this paper, we propose the CUR and generalized CUR decompositions of quaternion matrices and utilize the CUR decomposition of the quaternion matrix to solve the robust quaternion principal component analysis (RQPCA) problem. Through the discrete empirical interpolation method (DEIM) for the subset selection, we present the error analysis of the approximation of the CUR and the generalized CUR decompositions of quaternion matrices. The error bound depends on the conditioning of sampling submatrices. Next, we employ the alternating projection and adopt the CUR decomposition of the quaternion matrix to tackle the RQPCA. The non-convex algorithm runs fast and significantly reduces computational complexity. The performance advantages of our algorithms have been experimentally verified on artificial datasets. The RQCUR significantly affects color video background subtraction in the experiment.

Algebraic method for LU decomposition in commutative quaternion based on semi‐tensor product of matrices and application to strict image authentication

As a kind of commutative and associative four‐dimensional algebra, commutative quaternion has better applications in color image and signal processing than quaternion. Matrix decomposition is of great concern in the theoretical study and numerical calculation of commutative quaternion. Two kinds of disadvantages of commutative quaternion make the decomposition of commutative quaternion matrix extremely difficult. On one hand, commutative quaternion is not a kind of complete four dimensional division algebra because of the zero divisors. On the other hand, computing the inverse of commutative quaternion is very complicated. In this paper, the semi‐tensor product (STP) of matrices will be used to overcome the above two kinds of shortcomings. And we will propose a real structure‐preserving algorithm based on STP of matrices for commutative quaternion LU decomposition, which makes full use of high‐level operations, relation of operations between commutative quaternion matrices and their ‐representation matrices. Numerical experiments will be provided to demonstrate the efficiency of the real structure‐preserving algorithm based on STP of matrices. Meanwhile, we will apply the structure‐preserving algorithm for strict image authentication.

Algebraic method for LU decomposition of dual quaternion matrix and its corresponding structure-preserving algorithm

Algebraic method for LU decomposition of dual quaternion matrix and its corresponding structure-preserving algorithm

(R, S)-(Skew) Symmetric Solutions to Matrix Equation AXB = C over Quaternions

(R,S)-(skew) symmetric matrices have numerous applications in civil engineering, information theory, numerical analysis, etc. In this paper, we deal with the (R,S)-(skew) symmetric solutions to the quaternion matrix equation AXB=C. We use a real representation Aτ to obtain the necessary and sufficient conditions for AXB=C to have (R,S)-(skew) symmetric solutions and derive the solutions when it is consistent. We also derive the least-squares (R,S)-(skew) symmetric solution to the above matrix equation.

Systems of quaternionic linear matrix equations: solution, computation, algorithm, and applications

<p>In applied and computational mathematics, quaternions are fundamental in representing three-dimensional rotations. However, specific types of quaternionic linear matrix equations remain few explored. This study introduces new quaternionic linear matrix equations and their necessary and sufficient conditions for solvability. We employ a methodology involving lemmas and ranks of coefficient matrices to develop a novel algorithm. This algorithm is validated through numerical examples, showing its applications in advanced fields. In control theory, these equations are used for analyzing control systems, particularly for spacecraft attitude control in aerospace engineering and for control of arms in robotics. In quantum computing, quaternionic equations model quantum gates and transformations, which are important for algorithms and error correction, contributing to the development of fault-tolerant quantum computers. In signal processing, these equations enhance multidimensional signal filtering and noise reduction, with applications in color image processing and radar signal analysis. We extend our study to include cases of $ \eta $-Hermitian and i-Hermitian solutions. Our work represents an advancement in applied mathematics, providing computational methods for solving quaternionic matrix equations and expanding their practical applications.</p>