Search

In this study, we compute the unitarily invariant norms of the matrices, Az = (zizj)ni, j = 1, Bz = (zi - zj)ni, j = 1,and Cz = (zi/zj)n1j = 1, where zis are ith components of any complex sequence (zn). Moreover, we give some corollaries and numerical examples related to norms of these matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too. [ABSTRACT FROM AUTHOR]

Split-complex (hyperbolic) numbers are ordered pairs of real numbers, written in the form x + jy with j² = 1, used to describe the geometry of the Lorentzian plane. Since a null split-complex number does not have an inverse, some methods to calculate the exponential of complex matrices are not valid for split-complex matrices. In this paper, we examined the exponential of a 2 × 2 split-complex matrix in three cases: i. Δ = 0, ii. Δ ̸= 0 and Δ is not null split-complex number, iii. Δ ̸= 0 and Δ is a null split-complex number where Δ = (trA)2 - 4 detA. [ABSTRACT FROM AUTHOR]

Since complex interval matrix have many applications in different fields of science, in this paper interval complex matrix system as [W][Z] = [K] in which [W]; [K] are n × n known interval complex matrices and [Z] is n × n unknown interval complex matrix is studied. Using operations on interval complex numbers and matrices and defining a theorem, two auxiliary addition and subtraction complex systems are introduced and proved. Then, using the equality property of two complex numbers, the auxiliary interval complex systems are transformed to real crisp systems. Then the new system is solved and [Z] is achieved. Finally, some numerical examples are given to illustrate the applicability and ability of the proposed approach. [ABSTRACT FROM AUTHOR]

Matrix factorizations in dual number algebra, a hypercomplex number system, have been applied to kinematics, spatial mechanisms, and other fields recently. We develop an approach to identify spatiotemporal patterns in the brain such as traveling waves using the singular value decomposition (SVD) of dual matrices in this paper. Theoretically, we propose the compact dual singular value decomposition (CDSVD) of dual complex matrices with explicit expressions as well as a necessary and sufficient condition for its existence. Furthermore, based on the CDSVD, we report on the optimal solution to the best rank-k approximation under a newly defined quasi-metric in the dual complex number system. The CDSVD is also related to the dual Moore--Penrose generalized inverse. Numerically, comparisons with other available algorithms are conducted, which indicate less computational costs of our proposed CDSVD. In addition, the infinitesimal part of the CDSVD can identify the true rank of the original matrix from the noise-added matrix, but the classical SVD cannot. Next, we employ experiments on simulated time-series data and a road monitoring video to demonstrate the beneficial effect of the infinitesimal parts of dual matrices in spatiotemporal pattern identification. Finally, we apply this approach to the large-scale brain functional magnetic resonance imaging data, identify three kinds of traveling waves, and further validate the consistency between our analytical results and the current knowledge of cerebral cortex function. [ABSTRACT FROM AUTHOR]

The development of algorithmic differentiation (AD) tools focuses mostly on handling floating point types in the target language. Taping optimizations in these tools mostly focus on specific operations like matrix vector products. Aggregated types like std::complex are usually handled by specifying the AD type as a template argument. This approach provides exact results, but prevents the use of expression templates. If AD tools are extended and specialized such that aggregated types can be added to the expression framework, then this will result in reduced memory utilization and improve the timing for applications where aggregated types such as complex number or matrix vector operations are used. Such an integration requires a reformulation of the stored data per expression and a rework of the tape evaluation process. We will demonstrate the overheads on a synthetic benchmark and show the improvement when aggregated types are handled properly by the expression framework of the AD tool. [ABSTRACT FROM AUTHOR]

Let I n be the n × n identity matrix and J = 0 I n − I n 0 . A matrix A is called symplectic if A T J A = J. A symplectic matrix A is a commutator of symplectic involutions if A = X Y X − 1 Y − 1 , where X and Y are symplectic matrices satisfying X 2 = Y 2 = I. In this article, we give necessary and sufficient condition for a symplectic matrix over the complex number field to be expressed as a product of two commutators of symplectic involutions. [ABSTRACT FROM AUTHOR]

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same. [ABSTRACT FROM AUTHOR]

The Yang–Baxter–like matrix equation plays an important role in quantum group theory, knot theory and braid groups, which has received great attention from physicists and mathematicians. A difficult and important open problem is to find all the solutions of the Yang–Baxter–like matrix equation. As a matter of fact, it is difficult to find all of the solutions even when the coefficient matrix is a $ 3\times 3 $ 3 × 3 matrix. To the best of our knowledge, when the coefficient matrix is a $ 3\times 3 $ 3 × 3 diagonalizable complex matrix with three distinct nonzero eigenvalues, finding all the solutions of the Yang–Baxter–like matrix equation is still an open problem. In order to fill-in this gap, we first present a sufficient and necessary condition for the commuting solutions of the Yang–Baxter–like matrix equation, and give all the commuting solutions of the matrix equation. Second, with the help of a simplified matrix equation, we derive all the non–commuting solutions of the Yang–Baxter–like matrix equation by discussing whether the off–diagonal elements of the solutions are zero or not. [ABSTRACT FROM AUTHOR]

This paper focuses on algorithms for the truncated Takagi factorization of complex symmetric matrices. The problem is formulated as a Riemannian optimization problem on a complex Stiefel manifold and then is converted into a real Riemannian optimization problem on the intersection of the real Stiefel manifold and the quasi-symplectic set. The steepest descent, the Riemannian nonmonotone conjugate gradient, Newton, and hybrid methods are used for solving the problem and they are compared in their performance for the optimization task. Numerical experiments are provided to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient than other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices. [ABSTRACT FROM AUTHOR]

The split and hyperbolic (countercomplex) octonions are eight‐dimensional nonassociative algebras over the real numbers, which are in the form X=x0e0+∑j=17xjej, where em's 0≤m≤7 have different properties for them. The main purpose of this paper is to define the split‐type octonion and its matrix whose inputs are split‐type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split‐type octonions. Also, we show that every split‐type octonions can be represented by 2 × 2 real quaternion matrix and 4 × 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split‐type octonion matrix concept are given by using properties of  real quaternion matrices. Then, 8n × 8nreal matrix representations of split‐type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split‐type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split‐type octonion matrices to be special split‐type octonion matrices. We describe some special split‐type octonion matrices. Finally, oct‐determinant of split‐type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts. [ABSTRACT FROM AUTHOR]

This paper proposes a complex number representation method for dynamically recording robot positions and develops an optimization strategy for measuring and minimizing inter-robot distances in real-time. Based on this, the optimization strategy is transformed into finding the theoretical solution of the complex time-varying matrix equation by combining the leader–follower frameworks, and a complex zeroing neural dynamics (CZND) model is proposed for this objective. Meanwhile, the effectiveness of the proposed CZND model and the ideal trajectories of the follower robots are ensured and given through rigorous theoretical analyses. Subsequently, these theoretical findings are validated by two numerical experiments (four cases). In addition, the superiority of the CZND model for inter-robot management tasks is comparatively analyzed from three perspectives: initial positions, number of follower robots, and design parameters. Compared to conventional gradient-based neural network (CGNN) methods, the proposed CZND model, based on a dynamic error function (which is not required to be non-negative as in CGNN), demonstrates improved handling of the dynamic behavior of the system by utilizing the first-order time derivatives of matrices, resulting in lower residual errors in the implementation of inter-robot management. Specifically, for the same inter-robot management task, the steady-state error implemented by CZND is on the order of 1 0 − 4 , whereas for CGNN, it is on the order of 1 0 − 1 , indicating a significantly larger lagging error. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {M}_{n} $ M n be the set of $ n\times n $ n × n complex matrices, and for $ \varepsilon \gt 0 $ ε > 0 and $ A\in \mathcal {M}_{n} $ A ∈ M n , let $ \sigma _{\varepsilon }(A) $ σ ε (A) denote the $ \varepsilon - $ ε − pseudo spectrum of A. Maps Φ on $ \mathcal {M}_{n} $ M n which preserve the skew Lie product of matrices in a sense that \[ \sigma_{\varepsilon}(\Phi(A)\Phi(B) - \Phi(B)\Phi(A)^{*})=\sigma_{\varepsilon}(AB - BA^{*})\quad (A, B\in\mathcal{M}_{n}) \] σ ε (Φ (A) Φ (B) − Φ (B) Φ (A) ∗) = σ ε (AB − B A ∗) (A , B ∈ M n) are characterized, with no surjectivity assumption on them. Analogous description for the skew product on matrices is also noted. [ABSTRACT FROM AUTHOR]

It is well known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit i. The main aim of this paper is to demonstrate that a quaternion circulant matrix cannot be diagonalized by a discrete quaternion Fourier matrix with three imaginary units i, j, and k. Instead, a quaternion circulant matrix can be block-diagonalized into 1-by-1 block and 2-by-2 block matrices by permuted discrete quaternion Fourier transform matrix. With such a block-diagonalized form, the inverse of a quaternion circulant matrix can be determined efficiently similarly to the inverse of a complex circulant matrix. We make use of this block-diagonalized form to study quaternion tensor singular value decomposition of quaternion tensors where the entries are quaternion numbers. The applications, including computing the inverse of a quaternion circulant matrix and solving quaternion Toeplitz systems arising from linear prediction of quaternion signals, are employed to validate the efficiency of our proposed block-diagonalized results. [ABSTRACT FROM AUTHOR]

The multiparameter matrix pencil problem (MPP) is a generalization of the one-parameter MPP: given a set of r + 1, m x n complex matrices A0,...,Ar, with m ≥ n + r - 1, it is required to find all complex scalars λ0,...,λr, not all zero, such that the matrix pencil A(λ) = ... = 0λiAi loses column rank and the corresponding nonzero complex vector x such that A(λ)x = 0. We call the (r+1)-tuple λ0,...,λr an eigenvalue and the corresponding vector x an eigenvector. This problem is related to the well-known multiparameter eigenvalue problem except that there is only one pencil and, crucially, the matrices are not necessarily square. This paper uses our preliminary investigation in F. F. Alsubaie [H2 Optimal Model Reduction for Linear Dynamic Systems and the Solution of Multiparameter Matrix Pencil Problems, PhD thesis, Imperial College London, 2019], which presents a theoretical study of the multiparameter MPP and its applications in the H2 optimal model reduction problem, to give a full solution to the two-parameter MPP. First, an inflation process is implemented to show that the two-parameter MPP is equivalent to a set of three m² x n² simultaneous one-parameter MPPs. These problems are given in terms of Kronecker commutator operators (involving the original matrices) that exhibit several symmetries. These symmetries are analyzed and are then used to deflate the dimensions of the one-parameter MPPs to ... x ..., thus simplifying their numerical solution. In the case in which m = n + 1, it is shown that the two-parameter MPP has at least one solution and generically ... solutions, and furthermore that, under a rank assumption, the Kronecker determinant operators satisfy a commutativity property. This is then used to show that the two-parameter MPP is equivalent to a set of three simultaneous eigenvalue problems of dimension ... x .... A general solution algorithm is presented and numerical examples are given to outline the procedure of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper focuses on the solution of the reduced biquaternion equation AX+XB = C using the E-extra iteration method. By utilizing the complex decomposition of a reduced biquaternion matrix, we transform the equation into a complex matrix equation. Subsequently, we analyze the convergence of this method and provide guidelines for selecting optimal parameters. Finally, numerical examples are presented to demonstrate the efficacy of our algorithm. [ABSTRACT FROM AUTHOR]

In many stability problems, the investigation of pure imaginary roots for a polynomial family is very important. Given a pure imaginary complex number, the set of all images of uncertainty vectors is called the value set corresponding to this pure imaginary complex number. The question whether these sets contain the origin is very important from robust stability point of view of a polynomial family. Cutoff frequency guarantees the noninclusion of the origin to the value set for large frequencies. In this paper, we give a procedure for more strict estimation of cutoff frequency and applications of the obtained result to the constant inertia problem of a polynomial family. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be a fixed integer, and denote by M n the algebra of all n × n complex matrices. Fix also an integer k such that 1 ≤ k < n. We prove that if φ : M n → M n is a linear map which preserves the polynomial numerical radius of order k , there exist then a unitary n × n complex matrix U and a complex number ξ of modulus one such that either φ (T) = ξ U ⁎ T U for all T ∈ M n , or φ (T) = ξ U ⁎ T t U for all T ∈ M n. [ABSTRACT FROM AUTHOR]

Deep learning methods have gained significant popularity in the field of polarimetric synthetic aperture radar (PolSAR) image classification. These methods aim to extract high-level semantic features from the original PolSAR data to learn the polarimetric information. However, using only original data, these methods cannot learn multiple scattering features and complex structures for extremely heterogeneous terrain objects. In addition, deep learning methods always cause edge confusion due to the high-level features. To overcome these limitations, we propose a novel approach that combines a new double-channel convolutional neural network (CNN) with an edge-preserving Markov random field (MRF) model for PolSAR image classification, abbreviated to "DCCNN-MRF". Firstly, a double-channel convolution network (DCCNN) is developed to combine complex matrix data and multiple scattering features. The DCCNN consists of two subnetworks: a Wishart-based complex matrix network and a multi-feature network. The Wishart-based complex matrix network focuses on learning the statistical characteristics and channel correlation, and the multi-feature network is designed to learn high-level semantic features well. Then, a unified network framework is designed to fuse two kinds of weighted features in order to enhance advantageous features and reduce redundant ones. Finally, an edge-preserving MRF model is integrated with the DCCNN network. In the MRF model, a sketch map-based edge energy function is designed by defining an adaptive weighted neighborhood for edge pixels. Experiments were conducted on four real PolSAR datasets with different sensors and bands. The experimental results demonstrate the effectiveness of the proposed DCCNN-MRF method. [ABSTRACT FROM AUTHOR]

We consider sample covariance matrices where XN is a N × p real or complex matrix with i.i.d. entries with finite 12th moment and ΣN is a N × N positive definite matrix. In addition we assume that the spectral measure of ΣN almost surely converges to some limiting probability distribution as N → ∞ and p/N → γ > 0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type where I is the identity matrix, g is a bounded function and z is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse. [ABSTRACT FROM AUTHOR]

• Using the structural dynamic modification theory, the general damped system is considered a perturbation of the un-damped system, where the modification is defined by the damping matrix. C • The complexity of the mode shapes depends on the level of orthogonality of the normal mode shapes with respect to damping matrix. C. • The modal parameters of general damped systems (poles, complex modal masses, length of mode shapes) can be expressed as linear combination of the corresponding un-damped modal parameters. • If the mass density of the system is constant, the modal mass is always equal to the product between the total mass of the structure and the magnitude of the length squared, but with a phase to introduce a complex number. In this paper the modal mass is considered in the case of general damping where it is well-known that the modal mass becomes a complex valued quantity. The aim of this paper is to present a definition of the complex modal mass, which leads to an easier physical understanding of this modal parameter. It is demonstrated that based on the structural modification theory, the complex modal mass can be expressed as a linear combination of the un-damped modal masses, where a complex coefficient matrix can be defined that depends on the general damping matrix and on the mode shapes of the damped and the un-damped system. An important finding is that an apparent mass can be defined so that the modal mass is always equal to the product between the apparent mass and the length squared. The paper includes a 2-DOF example that illustrates the influence of the amount of general damping on modal quantities that are normally considered non-sensitive to damping. [ABSTRACT FROM AUTHOR]

By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the real representation of a complex matrix as a tool, the necessary and sufficient conditions for the convergence of the WRGI algorithm are determined. Also, some sufficient convergence conditions of the WRGI algorithm are presented. Moreover, the optimal step size and the corresponding optimal convergence factor of the WRGI algorithm are given. Lastly, some numerical examples are provided to demonstrate the effectiveness, feasibility and superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper focuses on the formation control of multi-robot systems with leader–follower network structure in directed topology to guide a system composed of multiple mobile robot agents to achieve global path navigation with a desired formation. A distributed linear formation control strategy based on the complex Laplacian matrix is employed, which enables the robot agents to converge into a similar formation of the desired formation, and the size and orientation of the formation are determined by the positions of two leaders. Additionally, in order to ensure that all robot agents in the formation move at a common velocity, the distributed control approach also includes a velocity consensus component. Based on the realization of similar formation control of a multi-robot system, the path navigation algorithm is combined with it to realize the global navigation of the system as a whole. Furthermore, a controller enabling the scalability of the formation size is introduced to enhance the overall maneuverability of the system in specific scenarios like narrow corridors. The simulation results demonstrate the feasibility of the proposed approach. [ABSTRACT FROM AUTHOR]

Full wavefront control by photonic components requires that the spatial phase modulation on an incoming optical beam ranges from 0 to 2π. Because of their radiative coupling to the environment, all optical components are intrinsically non‐Hermitian systems, often described by reflection and transmission matrices with complex eigenfrequencies. Here, it is shown that parity or time symmetry breaking—either explicit or spontaneous—moves the position of zero singularities of the reflection or transmission matrices from the real axis to the upper part of the complex frequency plane. A universal 0 to 2π‐phase gradient of an output channel as a function of the real frequency excitation is thus realized whenever the discontinuity branch bridging a zero and a pole, that is, a pair of singularities, is crossing the real axis. This basic understanding is applied to engineer electromagnetic fields at interfaces, including, but not limited to, metasurfaces. Non‐Hermitian topological features associated with exceptional degeneracies or branch cut crossing are shown to play a surprisingly pivotal role in the design of resonant photonic systems. [ABSTRACT FROM AUTHOR]

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to R I I -recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices. [ABSTRACT FROM AUTHOR]

Gaussian unitaries play a fundamental role in the field of continuous variables. In the general n mode, they may formulated by a second-order polynomial in the bosonic operators. Another important role related to Gaussian unitaries is played by the symplectic transformations in the phase space. The paper investigates the links between the two representations: the link from Hamiltonian to symplectic, governed by an exponential, and the link from symplectic to Hamiltonian, governed by a logarithm. Thus, an answer is given to the non-trivial question: which Hamiltonian produces a given symplectic representation? The complex instead of the traditional real symplectic representation is considered, with the advantage of getting compact and elegant relations. The application to the single, two, and three modes illustrates the theory. [ABSTRACT FROM AUTHOR]

This paper introduces trinion discrete cosine transform that can process color image holistically, it can be computed by combination of single-channel discrete cosine transform. Compared with quaternion discrete cosine transform, trinion discrete cosine transform is more efficient and compact to represent color image. Moreover, it is used for developing a robust color image encryption algorithm jointing with quantum logistic map and Josephus traversing. Firstly, color components are precoded into a trinion matrix, which is performed trinion discrete cosine transform. Then three parts of the transformed result are pairwisely combined into complex matrices and the synthesized spectrums satisfying symmetry are established. Followed by Josephus scrambling with variable steps on magnitudes, the ciphertext image can be acquired. The plaintext image can be ideally restored with the granted keys, where the average value of PSNR is close to 300.00 dB. Moreover, the proposed algorithm has high-level security. It also shows better resistance against Gaussian noise and shearing in comparison with some existing algorithms. [ABSTRACT FROM AUTHOR]

A new proof is presented of a theorem of L. Gurvits [LANL Unclassified Technical Report (2001), LAUR–01–2030], which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C(S^1)^{(n)} of n\times n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C(S^1)^{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) and C(S^1)_{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}), where \mathcal {H} is an arbitrary Hilbert space and C(S^1)_{(n)} is the operator system dual of C(S^1)^{(n)}. Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from \mathcal {B}(\mathcal {H}) when \mathcal {H} has infinite dimension. In particular, we prove that normal positive linear maps \psi on \mathcal {B}(\mathcal {H}) are partially completely positive in the sense that \psi ^{(n)}(x) is positive whenever x is a positive n\times n Toeplitz matrix with entries from \mathcal {B}(\mathcal {H}). We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando [Acta Sci. Math. (Szeged) 31 (1970), pp. 319–334] to universality. [ABSTRACT FROM AUTHOR]

In high-speed railways, the wireless channel and network topology change rapidly due to the high-speed movement of trains and the constant change of the location of communication equipment. The topology is affected by channel noise, making accurate channel estimation more difficult. Therefore, the way to obtain accurate channel state information (CSI) is the greatest challenge. In this paper, a two-stage channel-estimation method based on generative adversarial networks (cGAN) is proposed for MIMO-OFDM systems in high-mobility scenarios. The complex channel matrix is treated as an image, and the cGAN is trained against it to generate a more realistic channel image. In addition, the noise2noise (N2N) algorithm is used to denoise the pilot signal received by the base station to improve the estimation quality. Simulation experiments have shown the proposed N2N-cGAN algorithm has better robustness. In particular, the N2N-cGAN algorithm can be adapted to the case of fewer pilot sequences. [ABSTRACT FROM AUTHOR]

Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a slightly better estimate in the real case. Another one is a geometric application. Further results of a similar type are given for normal and almost symmetric matrices. [ABSTRACT FROM AUTHOR]

It is conjectured that four mutually unbiased bases in dimension 6 do not exist in quantum information. The conjecture is equivalent to the nonexistence of some three 6 x 6 complex Hadamard matrices (CHMs) with Schmidt rank at least 3. We investigate the 6 x 6 CHM U of Schmidt rank 3 containing two nonintersecting identical 3 x 3 submatrices V, i.e. .... We show that such U exists, V, W, X have rank 2 or 3, and they have rank 2 at the same time. We construct the analytical expressions of U when V is, respectively, of rank 2, unitary and normal. We apply our results to the conjecture by showing that U with some normal V is not one of the three 6 x 6 CHMs. [ABSTRACT FROM AUTHOR]

Let Ai, P, i = 1, ..., m, be n by n complex matrices such that each Ai is positive definite. It is shown, among other inequalities, that If 0 < aj ≤ sn(Aj), C m = ∑ j = 1 m (∑ i = 1 , i ≠ j m A i ) a j and P is Hermitian, then for every unitarily invariant norm ∥ |. ∥ | , ∥ | C m P + P C m ∥ | ≥ α ∥ | P ∥ | , where α = max 2 m − 1 , m 1 + min a 1 2 , ... , a m 2 . If s1(Aj) ≤ bj, ∑ i = 1 m b i = 1 and D m = ∑ j = 1 m (∑ i = 1 , i ≠ j m A i ) b j , then for every unitarily invariant norm ∥ |. ∥ | , 2 m P ≥ ∥ | D m P + P D m ∥ |. [ABSTRACT FROM AUTHOR]

Complex conference matrices have received considerable attention in the last few years due to their application in quantum information theory and in geometry. Various results and constructions are known for symmetric and Hermitian conference matrices. In this article, we deal mainly with constructions of complex skew-symmetric conference matrices. We classify all complex conference matrices up to order 5. Furthermore, we classify all complex symmetric, skew-symmetric and Hermitian conference matrices of order 6. [ABSTRACT FROM AUTHOR]

A complex square matrix is a ray pattern matrix if each of its nonzero entries has modulus 1. A ray pattern matrix naturally corresponds to a weighted-digraph. A ray pattern matrix A is ray nonsingular if for each entry-wise positive matrix K, A ∘ K is nonsingular. A random model of ray pattern matrices with order n is introduced, where a uniformly random ray pattern matrix B is defined to be the adjacency matrix of a simple random digraph D n , p whose arcs are weighted with i.i.d. random variables uniformly distributed over the unit circle in the complex plane. It is shown that p ∗ (n) = 1 n is a threshold function for the random matrix M = I−B to be ray nonsingular, which is the same as the threshold function of the appearance of giant strong components in D n , p . [ABSTRACT FROM AUTHOR]

We deal with the Roberts numerical radius orthogonality. In the case of 2 × 2 complex matrices, we give some necessary and sufficient conditions for the numerical range to be symmetric by employing the Roberts orthogonality with respect to the numerical radius. In addition, we present an interrelation between the Roberts orthogonality and the Birkhoff–James orthogonality with respect to the numerical radius for n × n complex matrices. Some other related results are also discussed. [ABSTRACT FROM AUTHOR]

Let Mn be the space of nxn complex matrices, and for ε>0 and A∈Mn, let σε(A) denote the ε -pseudo spectrum of A. Maps Φ on Mn which preserve the skew Jordan semi-triple product of matrices in a sense that σε(Φ(A)Φ(B)*Φ(A))=σε(AB*A)(A,B∈Mn) are characterized, with no surjectivity assumption on them. Analogous description is obtained for the skew Jordan product on matrices, and its variant of infinite dimension is also noted. [ABSTRACT FROM AUTHOR]

In this study, we introduce concepts of norms of elliptic complex matrices and derive the least squares solution, the pure imaginary least squares solution, and the pure real least squares solution with the least norm for the elliptic complex matrix equation AX=B by using the real representation of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic numbers are generalized form of complex and so real numbers. Thus, the obtained results extend, generalize, and complement some known least squares solutions results from the literature. [ABSTRACT FROM AUTHOR]

Given a complex idempotent matrix A, we derive simple, sufficient and necessary conditions for a matrix X being a nontrivial solution of the Yang-Baxter-like matrix equation A X A   =   X A X , discriminating commuting solutions from non-commuting ones. On this basis, we construct all the commuting solutions of the nonlinear matrix equation. [ABSTRACT FROM AUTHOR]

Dynamic complex matrix equation (DCME) is frequently encountered in the fields of mathematics and industry, and numerous recurrent neural network (RNN) models have been reported to effectively find the solution of DCME in no noise environment. However, noises are unavoidable in reality, and dynamic systems must be affected by noises. Thus, the invention of antinoise neural network models becomes increasingly important to address this issue. By introducing a new activation function (NAF), a robust zeroing neural network (RZNN) model for solving DCME in noisy-polluted environment is proposed and investigated in this paper. The robustness and convergence of the proposed RZNN model are proved by strict mathematical proof and verified by comparative numerical simulation results. Furthermore, the proposed RZNN model is applied to manipulator trajectory tracking control, and it completes the trajectory tracking task successfully, which further validates its practical applied prospects. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the solution of the autonomous q -difference system D q Y x = A Y q x with the initial condition Y 0 = Y 0 where A is a constant square complex matrix, D q is the Jackson q -derivative and 0 < q < 1 , is asymptotically stable if and only if ℜ λ < 0 for all λ ∈ σ A where σ A is the set of all eigenvalues of A (the spectrum of A). This results are exploited to provide the orthogonality property of the discrete q -Hermite matrix polynomials. [ABSTRACT FROM AUTHOR]

In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension n, are those linear maps that can be expressed as a convex combination of conjugations by n × n complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank N and the Choi rank r of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a Kraus representation of that channel. Most notably, we prove that the inequality N ≤ r 2 - r + 1 is satisfied for every mixed-unitary channel (as is the equality N = 2 when r = 2 ), and we exhibit the first known examples of mixed-unitary channels for which N > r . Specifically, we prove that there exist mixed-unitary channels having Choi rank d + 1 and mixed-unitary rank 2d for infinitely many positive integers d, including every prime power d. We also examine the mixed-unitary ranks of the mixed-unitary Werner–Holevo channels. [ABSTRACT FROM AUTHOR]

The invariant imbedding T-matrix (IIM T-matrix) model is regarded as one of the most promising models for calculating the scattering parameters of non-spherical particles. However, the IIM T-matrix model needs to be iterated along the radial direction when calculating the T-matrix, which involves complex calculations such as matrix inversion and multiplication. Therefore, how to improve its computational efficiency is an important problem to be solved. Focused on particles with N-fold symmetric geometry, this paper deduced the symmetry in the calculation process of the IIM T-matrix model, derived the block iteration scheme of the T-matrix, and contracted the IIM T-matrix program for particles with N-fold symmetric geometry. Discrete Dipole Approximation (DDA) and Geometrical Optics Approximation (IGOA) were employed to verify the accuracy of the improved IIM T-matrix model. The results show that the six phase matrix elements (P11, P12/P11, P22/P11, P33/P11, P34/P11 and P44/P11) calculated by our model are in good agreement with other models. The computational efficiency of the improved IIM T-matrix model was further investigated. As demonstrated by the results, the computational efficiency for the particles with N-fold symmetry improved by nearly 70% with the improvement of the symmetry of U matrix and T matrix. In conclusion, the improved model can remarkably reduce the calculation time while maintaining high accuracy. [ABSTRACT FROM AUTHOR]

As a first step of the study of the relation between the distribution of prime, reduced cycles of a graph and the limit theorem for the probability of a quantum walk on a graph, we present a trace formula with respect to the Grover matrix that is the transition matrix of the Grover walk on a graph. We define a zeta function with respect to the Grover matrix of a graph, and differentiate the logarithm of its zeta function of a regular graph. Furthermore, we obtain a trace formula with respect to the Grover matrix by a suitable complex integral of its logarithm differential. [ABSTRACT FROM AUTHOR]

Deep learning has powerful feature extraction abilities and has achieved promising results in polarimetric synthetic aperture radar (PolSAR) image classification. However, the labeled samples of PolSAR images are generally limited, which could lead to the overfitting of deep networks and the inefficiency of deep features. To overcome this problem, in this article, we propose a complex-valued enforcing population and lifetime sparsity (CV-EPLS) model to extract nonredundant sparse features from PolSAR images. CV-EPLS achieves unsupervised learning of sparse polarimetric features with limited and unlabeled samples, including amplitude and phase information in multiple polarimetric channels. Concretely, CV-EPLS defines an activation metric function to achieve strong population sparsity. Additionally, a grid search strategy is designed to ensure that activation items are evenly distributed among the sparse targets, thus forming strong lifetime sparsity. In this way, CV-EPLS constructs the complex sparse matrices and extracts discriminative sparse features in an unsupervised way, with the dependence of features being effectively reduced. Experimental results on PolSAR images demonstrate the effectiveness of CV-EPLS in the extraction of features and its application to image classification. [ABSTRACT FROM AUTHOR]

Specific features of the self-organization of various complex systems at the micro-, meso-, and macro-levels are investigated, taking environmental conditions into account. It is shown that the impact of the environment on a system in the process of self-organization can be modeled by relying on a four-digit measurement logic associated with the probabilities of type I and type II errors arising during comparisons. Harmonic self-organization stems from the quantization of actions, allowing measurement-action matrices to be transformed into quantum measurement matrices characterized by a parameter of the order of generalized golden ratios. Patterns of harmonic self-organization taking the form of complexes of elements in matrices are used to construct a quantum complex scale for classifying elements. Examples of manifestations of harmonic self-organization are considered. [ABSTRACT FROM AUTHOR]

In this paper, a two-sided, parallel Kogbetliantz-type algorithm for the hyperbolic singular value decomposition (HSVD) of real and complex square matrices is developed, with a single assumption that the input matrix, of order n, admits such a decomposition into the product of a unitary, a non-negative diagonal, and a J-unitary matrix, where J is a given diagonal matrix of positive and negative signs. When J = ±I, the proposed algorithm computes the ordinary SVD. The paper's most important contribution—a derivation of formulas for the HSVD of 2 × 2 matrices—is presented first, followed by the details of their implementation in floating-point arithmetic. Next, the effects of the hyperbolic transformations on the columns of the iteration matrix are discussed. These effects then guide a redesign of the dynamic pivot ordering, being already a well-established pivot strategy for the ordinary Kogbetliantz algorithm, for the general, n × n HSVD. A heuristic but sound convergence criterion is then proposed, which contributes to high accuracy demonstrated in the numerical testing results. Such a J-Kogbetliantz algorithm as presented here is intrinsically slow, but is nevertheless usable for matrices of small orders. [ABSTRACT FROM AUTHOR]