Quaternion Matrix Research Articles

Minimax Principle for Eigenvalues of Dual Quaternion Hermitian Matrices and Generalized Inverses of Dual Quaternion Matrices

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency concepts for a set of dual quaternion vectors, and study some related basic properties for dual quaternion vectors and dual quaternion matrices. We present a minimax principle for eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an inequality for singular values of dual quaternion matrices. Finally, we introduce the concept of generalized inverses of dual quaternion matrices, and present necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix.

A coupled quaternion matrix equations with applications

A coupled quaternion matrix equations with applications

The solutions of the quaternion matrix equation AXε + BXδ = 0

In this paper, we discuss the quaternion matrix equation AXε+BXδ=0, where ε∈{I,C}, δ∈{†,⁎} and I, C, †, ⁎ denote the identity, involutive automorphism, involutive anti-automorphism and transpose, involutive automorphism and anti-automorphism and transpose, respectively. Firstly, we transform the given matrix equation into the matrix equation A˜Yε+B˜Yδ=0 with complex coefficient matrices and quaternion target matrix Y by utilizing the regularity of (A,B), where A˜=PAQ and B˜=PBQ are two complex matrices with P,Q being two invertible quaternion matrices. Secondly, we show that the solution can be obtained in terms of Q, the Kronecker canonical form of (A˜,B˜) and one of the two invertible quaternion matrices which transform (A˜,B˜) into its Kronecker canonical form. Meanwhile, we also decouple the transformed equation into some systems of small-scale equations in terms of Kronecker canonical form of (A˜,B˜). Thirdly, we determine the dimension of the solution space of the equation in terms of the sizes of the blocks arising in the Kronecker canonical form. Finally, we give the necessary and sufficient condition for the existence of the unique solution.

On singular value decomposition and generalized inverse of a commutative quaternion matrix and applications

By means of a complex representation of a commutative quaternion matrix, the singular value decomposition and the generalized inverse problems of a commutative quaternion matrix are studied, and the corresponding theorems and algorithms are given. In addition, based on the singular value decomposition and generalized inverse of a commutative quaternion matrix, the numerical experiments for solving the least squares problem and the color image watermarking problem are given. Numerical experiments illustrate the effectiveness and reliability of the proposed algorithms.

W-MPCEP–N-CEPMP-solutions to quaternion matrix equations with constrains

W-MPCEP–N-CEPMP-solutions to quaternion matrix equations with constrains

The General Solution to a System of Linear Coupled Quaternion Matrix Equations with an Application

The General Solution to a System of Linear Coupled Quaternion Matrix Equations with an Application

Quaternion algebra approach to nonlinear Schrödinger equations with nonvanishing boundary conditions

In this article we apply quaternionic linear algebra and quaternionic linear system theory to develop the inverse scattering transform theory for the nonlinear Schrödinger equation with nonvanishing boundary conditions. We also determine its soliton solutions by using triplets of quaternionic matrices.

Error-Covariance Reset in the Multiplicative Extended Kalman Filter for Attitude Estimation

This paper presents a study of the reset step in the multiplicative extended Kalman filter (MEKF). This filter is widely used for spacecraft attitude estimation, which typically involves estimating the attitude and gyro drift in real time using external sensors such as star trackers. The basic idea of the MEKF is to use the quaternion or direction-cosine matrix as the “global” attitude parameterization and a three-component state vector for the “local” parameterization of attitude errors. The true attitude is expressed as the product of the error attitude and the estimate rather than as the sum of the error and the estimate. The reset operation moves the local error to the global variable. This reset does not add new information, but it changes the reference frame for the attitude error covariance. This results in an error-covariance reset that is very different from the measurement update of the error covariance in the MEKF. The effects of using an error-covariance reset in the MEKF are analyzed in this work. The results from this work can be applied to any application involving attitude estimation as part of its process, such as inertial navigation.

On singular value decomposition for split quaternion matrices and applications in split quaternionic mechanics

In the theoretical studies and numerical computations of split quaternionic quantum mechanics, the singular value decomposition theorem of split quaternion matrices plays an essential role. In general the problem of singular value decomposition over split quaternion algebra has hitherto remained tangential for split quaternion matrices. In this paper, by means of a real representation matrix of a split quaternion matrix, the singular value decomposition of split quaternion matrices is studied, the singular value decomposition theorem and the corresponding algorithm for split quaternion matrices are given. Finally, three applications for solving the split quaternion least squares problem and the wave separation problem are given based on the singular value decomposition over split quaternion algebra.

A complex structure-preserving algorithm for computing the singular value decomposition of a quaternion matrix and its applications

A complex structure-preserving algorithm for computing the singular value decomposition of a quaternion matrix and its applications

Towards Higher-Order Zeroing Neural Networks for Calculating Quaternion Matrix Inverse with Application to Robotic Motion Tracking

The efficient solution of the time-varying quaternion matrix inverse (TVQ-INV) is a challenging but crucial topic due to the significance of quaternions in many disciplines, including physics, engineering, and computer science. The main goal of this research is to employ the higher-order zeroing neural network (HZNN) strategy to address the TVQ-INV problem. HZNN is a family of zeroing neural network models that correlates to the hyperpower family of iterative methods with adjustable convergence order. Particularly, three novel HZNN models are created in order to solve the TVQ-INV both directly in the quaternion domain and indirectly in the complex and real domains. The noise-handling version of these models is also presented, and the performance of these models under various types of noises is theoretically and numerically tested. The effectiveness and practicality of these models are further supported by their use in robotic motion tracking. According to the principal results, each of these six models can solve the TVQ-INV effectively, and the HZNN strategy offers a faster convergence rate than the conventional zeroing neural network strategy.

Near-infrared spectroscopy analysis of compound fertilizer based on GAF and quaternion convolution neural network

This paper combines near infrared spectroscopy with deep learning theory to propose a rapid identification method for compound fertilizer based on gramian angular field (GAF) image coding and quaternion convolutional neural network (QCNN). First, the near-infrared (NIR) spectra of 200 samples of two types of compound with and without polyglutamic acid (γ-PGA) were collected and pretreated by multivariate scattering correction(MSC) and first derivative methods. Then, the one-dimensional (1D) NIR spectra were transformed into two-dimensional (2D) images by GAF coding method to express the spectral information more intuitively. Finally, three images of Gramian angular difference field (GADF), Gramian angular summation field (GASF) and their average images were formed into a pure quaternion matrix for parallel representation and were analyzed by the designed QCNN. The peak features and minor features were mined based on the automatic learning function of multi-layer structure of QCNN to establish a high-performance, qualitative model for identifying compound fertilizer. The experimental results show that the classification accuracy, sensitivity, and specificity of the identification model were 95.84%, 95.96% and 94.25%, respectively. Compared with the classification results based on traditional partial least square discrimination analysis(PLS-DA), principal component analysis combing with support vector machine classification(PCA-SVM), 1D convolution neural network (1DCNN) and GAF-CNN methods, the classification accuracy and adaptability of the model based on the proposed GAF-QCNN method have been significantly improved. The combination of GAF image of NIR spectra and QCNN developed in this study provides a novel approach for the intelligent modeling algorithm of NIR spectroscopy technology.

Mutually Unbiased Measurements, Hadamard Matrices, and Superdense Coding

Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert spaces, and the corresponding rank-1 projective measurements are ubiquitous in quantum information theory. In this work, we study a recently introduced generalisation of MUBs called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mutually unbiased measurements</i> (MUMs). These measurements inherit the essential property of complementarity from MUBs, but the Hilbert space dimension is no longer required to match the number of outcomes. This operational complementarity property renders MUMs highly useful for device-independent quantum information processing. It has been shown that MUMs are strictly more general than MUBs. In this work we provide a complete proof of the characterisation of MUMs that are direct sums of MUBs. We then construct new examples of MUMs that are not direct sums of MUBs. A crucial technical tool for this construction is a correspondence with quaternionic Hadamard matrices, which allows us to map known examples of such matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that–in stark contrast with MUBs–the number of MUMs for a fixed outcome number is unbounded. Next, we focus on the use of MUMs in quantum communication. We demonstrate how any pair of MUMs with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> outcomes defines a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -dimensional superdense coding protocol. Using MUMs that are not direct sums of MUBs, we disprove a recent conjecture due to Nayak and Yuen on the rigidity of superdense coding, for infinitely many dimensions. The superdense coding protocols arising in the refutation reveal how shared entanglement may be used in a manner heretofore unknown.

Controllability and observability for linear quaternion-valued impulsive differential equations

In this paper, the controllability and observability for linear quaternion-valued impulsive differential equations (QIDEs) are investigated. We mainly establish some sufficient and necessary conditions for state controllability and state observability of linear QIDEs. By virtue of the isomorphism between quaternion vector space and complex variables space as well as the adjoint matrix of quaternion matrix, all the theoretical results in the sense of complex-valued and quaternion-valued are equivalent to each other. Finally, the validity of theoretical results acquired is indicated by two instances shown.

Probabilistic quaternion collaborative representation and its application to robust color face identification

Quaternion representation (QR) has been shown to be an effective tool for image processing by modelling color images as quaternion matrices. However, previous QR based classification (QRC) methods overlook the labels of training data and may lead to sub-optimal results. In this article, we propose a Probabilistic Quaternion Collaborative Representation (ProQCR) method based on maximum likelihood estimation with application to color face identification. The proposed ProQCR method has the following merits: (1) Full supervision: In the representation and classification processes, the ProQCR method exploits the label information of training data, in contrast to prior methods using such supervision information only in the classification step. (2) Multi-channel integrality: ProQCR represents multiple color channels of each test color image as a quaternion linear combination of training data in a holistic manner. (3) Robustness: Equipped with the quaternion Huber loss function, ProQCR is robust against gross corruption in the test data. Experiments on benchmark databases demonstrate the effectiveness and robustness of ProQCR for color face identification.

The consistency and the general common solution to some quaternion matrix equations

The consistency and the general common solution to some quaternion matrix equations

Quaternion Matrix Factorization for Low-Rank Quaternion Matrix Completion

The main aim of this paper is to study quaternion matrix factorization for low-rank quaternion matrix completion and its applications in color image processing. For the real-world color images, we proposed a novel model called low-rank quaternion matrix completion (LRQC), which adds total variation and Tikhonov regularization to the factor quaternion matrices to preserve the spatial/temporal smoothness. Moreover, a proximal alternating minimization (PAM) algorithm was proposed to tackle the corresponding optimal problem. Numerical results on color images indicate the advantages of our method.

The $$\eta $$-(anti-)Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation

The $$\eta $$-(anti-)Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation

Characterizations of Unit Darboux Ruled Surface with Quaternions

This paper presents a quaternionic approach to generating and characterizing the ruled surface drawn by the unit Darboux vector. The study derives the Darboux frame of the surface and relates it to the Frenet frame of the base curve. Moreover, it obtains the quaternionic shape operator and its matrix representation using the normal and geodesic curvatures to provide a more detailed analysis. To illustrate the concepts discussed, the paper offers a clear example that will help readers better understand the concepts and showcases the quaternionic shape operator, Gauss curvature, mean curvature, and rotation matrix. Finally, it emphasizes the need for further research on this topic.

New insight into quaternions and their matrices

This paper aims to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and their algebraic structures are examined. Several matrix representations and computational results are introduced. An alternative approach for a generalized quaternion matrix with elliptic number entries has been developed as a crucial part.