Quaternion Matrix Research Articles

Special least squares solutions of the quaternion matrix equation [formula omitted

In this paper, by using the real representations of quaternion matrices, the particular structure of the real representations of quaternion matrices, the Kronecker product of matrices and the Moore–Penrose generalized inverse, we obtain the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation AXB+CXD=E, respectively. Our resulting formulas only involve real matrices, and therefore are simpler than those reported in Yuan (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of quaternion matrices, and therefore are very efficient and portable. Numerical examples are provided to illustrate the efficiency of our algorithms.

The quaternionic second weighted zeta function of a graph and the Study determinant

We establish a generalization of the second weighted zeta function of a graph to the case of quaternions. For an arc-weighted graph whose weights are quaternions, we define the second weighted zeta function by using the Study determinant that is a quaternionic determinant for quaternionic matrices defined by Study. This definition is regarded as a quaternionic analogue of the determinant expression of Hashimoto type for the Ihara zeta function of a graph. We derive the Study determinant expression of Bass type and the Euler product for the quaternionic second weighted zeta function.

Sparse representation based on vector extension of reduced quaternion matrix for multiscale image denoising

Sparse representations of multi-channel signals have drawn considerable interest in recent years. In this study, a new vector-valued sparse representation model is proposed for colour images using reduced quaternion matrix (RQM). The colour image is described as a RQM by the proposed model. In the dictionary training state, k-means clustering RQM value decomposition is proposed which makes sparse basis selection in quaternion space. Then, a reduced quaternion-based orthogonal matching pursuit algorithm is presented in the sparse coding stage. To demonstrate the effectiveness of the proposed sparse representation model, the authors apply the model to common colour image processing problem-colour image denoising. The proposed model is compared with other sparse models for colour image denoising in terms of visual quality and peak signal-to-noise ratio. The experimental results indicate that the proposed image sparse model is competitive with other sparse models.

Quaternionic Approach of Canal Surfaces Constructed by Some New Ideas

A canal surface is a surface constructed as the envelope of a family of spheres with the parametric (non constant radii) radii r(s) and a space curve $${\alpha (s)}$$ called its center. If the radii of the generating spheres are constant r then the canal surface is called tubular surface such as right circular cylinder, torus, right circular cone, surface of revolution. The quaternions are number systems generalized by the complex numbers. Irish mathematician Hamilton (Lond Edinb Dublin Philos Mag J Sci 25(3):489–495, 1844) discovered them in 1843 and applied to mechanics in three-dimensional space. Many laws of curves and surfaces in differential geometry used quaternions such as quaternionic helices (Coken and Tuna in Appl Math Comput 155:373–389, 2004; Gok et al. in Appl Clifford Algebras 21:707–719, 2011; Kahraman et al. in Appl Math Comput 218:6391–6400, 2012) and canal surfaces (Aslan and Yayli in Adv Appl Clifford Algebras, 2015; Babaarslan and Yayli in ISRN Geom 2012:Article ID 126358, p 8, 2012). This paper has two purposes. The first purpose is to form canal surfaces whose centers are spherical indicatrices of a spatial curve with a new idea in terms of alternative moving frame $${\{N,C,W\}}$$ The second purpose is to obtain canal surfaces by using the quaternion product and matrix representation. Moreover, we give some related examples with their figures.

Quaternionic approach to equiform kinematics and line-elements of Euclidean 4-space and 3-space

We extend the quaternionic kinematic mapping of Euclidean displacements of Euclidean 4-space E4 to the group of equiform transformations S(4). As a consequence the equiform motions of basic elements (points, oriented lines, oriented planes, oriented hyperplanes) of E4 can be written compactly in terms of 2×2 quaternionic matrices. This representation is extended to oriented line-elements of E4 and to instantaneous screws of S(4), for which a classification (incl. corresponding normal forms) is given. Based on this preparatory work we study the relation between instantaneous equiform motions and the geometry of line-elements (path normal-elements, path tangent-elements) in E4. Finally, we show that the line-elements of projective 3-space can be mapped bijectively on the Segre variety Σ3,2.

Centrohermitian and Skew-Centrohermitian Solutions to the Minimum Residual and Matrix Nearness Problems of the Quaternion Matrix Equation $${(AXB,DXE) = (C,F)}$$ ( A X B , D X E ) = ( C , F )

It is established that the precise solutions on the minimum residual and matrix nearness problems of the quaternion matrix equation \({(AXB,DXE)=(C,F)}\) for centrohermitian and skew-centrohermitian matrices, where X is an unknown quaternion matrix and A, B, C, D, E, and F are known quaternion matrices of suitable sizes. Moreover, an algorithm to get the solutions of the problems considered is provided, and a numerical example is also given to exemplify the results.

O(∞)- and Sp(∞)-Invariant Ergodic Measures on the Spaces of Infinite Antisymmetric and Quaternionic Antihermitian Matrices

The ergodic measures for the actions of the infinite orthogonal group O(∞) and the infinite symplectic group Sp(∞) by conjugations on the spaces of infinite real antisymmetric and infinite quaternionic antihermitian matrices are classified, with the use of the Vershik–Olshanski “ergodic method.”

The General Solution to a System of Eight Quaternion Matrix Equations with Applications

This paper studies some necessary and sufficient conditions for the existence of a solution and gives an expression of the general solution to the system of eight linear quaternion matrix equations. As an application, necessary and sufficient conditions are given for the system of certain matrix equations to have a symmetric solution. Note that in some of the mentioned conditions we use rank equalities. In addition, some numerical examples are given.

Real structure-preserving algorithms of Householder based transformations for quaternion matrices

In this paper, we survey three different forms of Householder based transformations for quaternion matrices in the literature, and propose a new form of quaternion Householder based transformation. We propose real structure-preserving algorithms of these Householder based transformations, which make the procedure computationally more flexible and efficient. We compare the computation counts and assignment numbers of these algorithms. We also compare the effectiveness of these real structure-preserving algorithms applying to the quaternion QRD and the quaternion SVD.All these four real structure-preserving algorithms are more efficient, comparing to the algorithms which apply Quaternion Toolbox using quaternion arithmetics, or algorithms which directly performs real Householder transformations on the real representation of a quaternion matrix. Among these four real structure-preserving algorithms, the most efficient ones are based on quaternion Householder reflection, and new proposed Householder based transformation.

Some results on the k-numerical range of quaternion matrices

Let n and k be two positive integers and . In this paper, the connectivity of the k-numerical range of n-square quaternion matrices is investigated. Also, the quaternion matrices whose higher numerical ranges are singleton or a line segment are characterized. Moreover, some algebraic and geometrical properties of higher numerical ranges of skew-Hermitian quaternion matrices are investigated. The n-numerical range of n-square quaternion matrices is also studied.

An efficient solver for large structured eigenvalue problems in relativistic quantum chemistry

ABSTRACTWe report an efficient program for computing the eigenvalues and symmetry-adapted eigenvectors of very large quaternionic (or Hermitian skew-Hamiltonian) matrices, using which structure-preserving diagonalisation of matrices of dimension N > 10, 000 is now routine on a single computer node. Such matrices appear frequently in relativistic quantum chemistry owing to the time-reversal symmetry. The implementation is based on a blocked version of the Paige–Van Loan algorithm, which allows us to use the Level 3 BLAS subroutines for most of the computations. Taking advantage of the symmetry, the program is faster by up to a factor of 2 than state-of-the-art implementations of complex Hermitian diagonalisation; diagonalising a 12, 800 × 12, 800 matrix took 42.8 (9.5) and 85.6 (12.6) minutes with 1 CPU core (16 CPU cores) using our symmetry-adapted solver and Intel Math Kernel Library's ZHEEV that is not structure-preserving, respectively. The source code is publicly available under the FreeBSD licence.

Hybrid Color and Grayscale Images Encryption Scheme Based on Quaternion Hartley Transform and Logistic Map in Gyrator Domain

A hybrid color and grayscale images encryption scheme based on the quaternion Hartley transform (QHT), the two-dimensional (2D) logistic map, the double random phase encoding (DRPE) in gyrator transform (GT) domain and the three-step phase-shifting interferometry (PSI) is presented. First, we propose a new color image processing tool termed as the quaternion Hartley transform, and we develop an efficient method to calculate the QHT of a quaternion matrix. In the presented encryption scheme, the original color and grayscale images are represented by quaternion algebra and processed holistically in a vector manner using QHT. To enhance the security level, a 2D logistic map-based scrambling technique is designed to permute the complex amplitude, which is formed by the components of the QHT-transformed original images. Subsequently, the scrambled data is encoded by the GT-based DRPE system. For the convenience of storage and transmission, the resulting encrypted signal is recorded as the real-valued interferograms using three-step PSI. The parameters of the scrambling method, the GT orders and the two random phase masks form the keys for decryption of the secret images. Simulation results demonstrate that the proposed scheme has high security level and certain robustness against data loss, noise disturbance and some attacks such as chosen plaintext attack.

Bounds for Eigenvalues of Matrix Polynomials Over Quaternion Division Algebra

Localization theorems are discussed for the left and right eigenvalues of block quaternionic matrices. Basic definitions of the left and right eigenvalues of quaternionic matrices are extended to quaternionic matrix polynomials. Furthermore, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where \({p = 1, 2, \infty, F}\). The above generalizes the bounds on the absolute values of the eigenvalues of complex matrix polynomials, which give sharper bounds to the bounds developed in [LAA, 358, pp. 5–22 2003] for the case of 1, 2, and \({\infty}\) matrix norms.

Analogies between random matrix ensembles and the one-component plasma in two-dimensions

The eigenvalue PDF for some well known classes of non-Hermitian random matrices — the complex Ginibre ensemble for example — can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We address this theme in a systematic fashion, identifying the plasma system for the Ginibre ensemble of non-Hermitian Gaussian random matrices G, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix G1−1G2, and the ensemble formed by truncating unitary matrices, as well as for products of such matrices. We do this when each has either real, complex or real quaternion elements. One consequence of this analogy is that the leading form of the eigenvalue density follows as a corollary. Another is that the eigenvalue correlations must obey sum rules known to characterise the plasma system, and this leads us to an exhibit of an integral identity satisfied by the two-particle correlation for real quaternion matrices in the neighbourhood of the real axis. Further random matrix ensembles investigated from this viewpoint are self dual non-Hermitian matrices, in which a previous study has related to the one-component plasma system in a disk at inverse temperature β=4, and the ensemble formed by the single row and column of quaternion elements from a member of the circular symplectic ensemble.

An improved feature extraction-based colour image matching method using quaternion matrix

Specific to the problem of feature description for colour image, a novel speeded-up robust features SURF descriptor of colour image-based quaternion CSURF-Q is proposed in this paper. Firstly, in order to represent the colour information of colour image effectively, the colour information among three channels is expressed by the three imaginary parts of quaternion. This can transform colour image to a pure quaternion matrix, which simplifies the representation process of image colour information. Secondly, the SURF features among three channels of colour image are extracted by using quaternion matrix. Because of the rotation invariance of quaternion norm, feature description vectors are generated as the form of quaternion norm. Finally, comparing CSURF-Q with SURF, FAIR-SURF, CSIFT, SIFT-CCH, colour-SURF and CHSIFT, the experiments prove that, when image appears affine changes, the correct matching of feature point pairs and time complexity are superior.

Matrices over hyperbolic split quaternions

In this paper, we present some important properties of matrices over hyperbolic split quaternions. We examine hyperbolic split quaternion matrices by their split quaternion matrix representation.

Explicit Determinantal Representation Formulas ofW-Weighted Drazin Inverse Solutions of Some Matrix Equations over the Quaternion Skew Field

By using determinantal representations of theW-weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of theW-weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equationsWAWX=D,XWBW=D, andW1AW1XW2BW2=D.

Group Symmetric Robust Covariance Estimation

In this paper, we consider Tyler's robust covariance M-estimator under group symmetry constraints. We assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group, referred to as group symmetry. Examples of group symmetric structures include circulant, perHermitian, and proper quaternion matrices. We introduce a group symmetric version of Tyler's estimator (STyler) and provide an iterative fixed point algorithm to compute it. The classical results claim that at least n=p+1 sample points in general position are necessary to ensure the existence and uniqueness of Tyler's estimator, where p is the ambient dimension. We show that the STyler requires significantly less samples. In some groups, even two samples are enough to guarantee its existence and uniqueness. In addition, in the case of elliptical populations, we provide high probability bounds on the error of the STyler. These, too, quantify the advantage of exploiting the symmetry structure. Finally, these theoretical results are supported by numerical simulations.

New results on condensed Cramer’s rule for the general solution to some restricted quaternion matrix equations

In this paper, we derive some condensed Cramer’s rules for the general solution, the least squares solution and the least norm solution to some restricted quaternion matrix equations, respectively. The findings of this paper extend some known results in the literature.

The discrete-time quaternionic quantum walk on a graph

Recently, the quaternionic quantum walk was formulated by the first author as a generalization of discrete-time quantum walks. We deal with the right eigenvalue problem of quaternionic matrices in order to study spectra of the transition matrix of a quaternionic quantum walk. The way to obtain all the right eigenvalues of a quaternionic matrix is given. From the unitary condition on the transition matrix of a quaternionic quantum walk, we deduce some remarkable properties of it. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using those of the corresponding weighted matrix. In addition, we give some examples of quaternionic quantum walks and their right eigenvalues.