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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. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study for the first time the concept of square real matrices with symbolic 2-plithogenic split-complex entries. Many of their algebraic properties will be discussed and handled, where we find the formula of computing inverses, exponents, and powers of these matrices by building a ring isomorphism between the ring of split-complex symbolic 2-plithogenic matrices and the direct product of the symbolic 2-plithogenic matrices with itself. Also, we give the interested reader many related examples to clarify the validity of our work. [ABSTRACT FROM AUTHOR]

Classical matrix theory for real, complex and hypercomplex numbers is a well-known concept. Is it possible to construct matrix theory over dual-generalized complex (DGC) matrices? The answer to this question is given in this paper. The paper is constructed as follows. Firstly, the fundamental concepts for DGC matrices are introduced and DGC special matrices are defined. Then, theoretical results related to eigenvalues/eigenvectors are obtained and universal similarity factorization equality (USFE) regarding to the dual fundamental matrix are presented. Also, spectral theorems for Hermitian and unitary matrices are introduced. Finally, due to the importance of unitary matrices, a method for finding a DGC unitary matrix is stated and examples for spectral theorem are given. [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]

We present a generalization of the pseudoinverse operation to pairs of matrices, as opposed to single matrices alone. We note the fact that the Singular Value Decomposition can be used to compute the ordinary Moore-Penrose pseudoinverse. We present an analogue of the Singular Value Decomposition for pairs of matrices, which we show is inadequate for our purposes. We then present a more sophisticated analogue of the SVD which includes features of the Jordan Normal Form, which we show is adequate for our purposes. This analogue of the SVD, which we call the Jordan SVD, was already presented in a previous paper by us called 'Matrix decompositions over the double numbers'. We adopt the idea presented in that same paper that a pair of matrices is actually a single matrix over the double-complex number system. [ABSTRACT FROM AUTHOR]

The solution of a complex-valued matrix pseudoinverse is one of the key steps in various science and engineering fields. Owing to its important roles, researchers had put forward many related algorithms. With the development of research, a time-varying matrix pseudoinverse received more attention than a time-invarying one, as we know that a zeroing neural network (ZNN) is an efficient method to calculate a pseudoinverse of a complex-valued time-varying matrix. Due to the initial ZNN (IZNN) and its extensions lacking a mechanism to deal with both convergence and robustness, that is, most existing research on ZNN models only studied the convergence and robustness, respectively. In order to simultaneously improve the double features (i.e., convergence and robustness) of ZNN in solving a complex-valued time-varying pseudoinverse, this paper puts forward a double features ZNN (DFZNN) model by adopting a specially designed time-varying parameter and a novel nonlinear activation function. Moreover, two nonlinear activation types of complex number are investigated. The global convergence, predefined time convergence, and robustness are proven in theory, and the upper bound of the predefined convergence time is formulated exactly. The results of the numerical simulation verify the theoretical proof, in contrast to the existing complex-valued ZNN models, the DFZNN model has shorter predefined convergence time in a zero noise state, and enhances robustness in different noise states. Both the theoretical and the empirical results show that the DFZNN model has better ability in solving a time-varying complex-valued matrix pseudoinverse. Finally, the proposed DFZNN model is used to track the trajectory of a manipulator, which further verifies the reliability of the model. [ABSTRACT FROM AUTHOR]

In the paper, we study and compare relative (k, n) Valiron defect with the relative Nevanlinna defect for meromorphic function where k and n are both non negative integers on annuli. The results we proved are as follows 1. Let f(z) be a transcendental or admissible meromorphic function of finite order in A(R0), where 1 < R0 ≤ +∞ and Σ a≠∞ δ0(a, f) + δ0(∞, f) = 2. Then... 2. Let f(z) be a transcendental or admissible meromorphic function of finite order in A(R0), where 1 < R0 ≤ +∞ such that m0(r, f) = S(r, f). If a, b and c are three distinct complex numbers, then for any two positive integer k and n 3Rδ(0) 0(n)(a, f) + 2Rδ(0) 0(n)(b, f) + 3Rδ(0)0(n)(c, f) + 5RΔ(k) 0(n)(∞, f) ≤ 5RΔ(0) 0(n)(∞, f) + 5RΔ(k) 0(n) ≤ +∞ such that m0(r, f) = S(r, f). If a, b and c are three distinct complex numbers, then for any two positive integer k and n Rδ(0) 0(n)(0, f) +R Δ(k) 0(n)(∞, f) +R0), where 1 < R0 ≤ +∞ such that m0(r, f) = S(r, f). If a, b and c are three distinct complex numbers, then for any two positive integer k and n Rδ(0) 0(n)(0, f) +R Δ(k) 0(n)(∞, f) +Rδ(0)0(n)(c, f) ≤R Δ(0) 0(n)(∞, f) + 2RΔ(k) ≤ +∞ such that m0(r, f) = S(r, f). If a and d are two distinct complex numbers, then for any two positive integer k and p with 0 ≤ k ≤ p Rδ(0) 0(n)(d, f) +R Δ(p) 0(n)(∞, f) +R δ(k) 0(n)(a, f) ≤R Δ(k) 0(n)(∞, f) +R Δ(p) 0(n)(0, f) +R Δ(k)0(n)(0, f), where n is any positive integer. 5.Let f(z) be a transcendental or admissible meromorphic function of finite order in A(R0(n)(0, f). 4. Let f(z) be a transcendental or admissible meromorphic function of finite order in A(R0 ≤ +∞. Then for any two positive integers k and n, RΔ(0) 0(n)(∞, f) +R Δ(k) 0(n)(0, f) ≥R δ(0) 0(n)(0, f) +R δ(0) 0(n)(a, f) +R Δ(k) 0(n)(∞, f), where a is any non zero complex number. [ABSTRACT FROM AUTHOR]0 ≤ +∞ such that m0(r, f) = S(r, f). If a and d are two distinct complex numbers, then for any two positive integer k and p with 0 ≤ k ≤ p Rδ(0) 0(n)(d, f) +R Δ(p) 0(n)(∞, f) +R δ(k) 0(n)(a, f) ≤R Δ(k) 0(n)(∞, f) +R Δ(p) 0(n)(0, f) +R Δ(k)0(n)(0, f), where n is any positive integer. 5.Let f(z) be a transcendental or admissible meromorphic function of finite order in A(R0), where 1 < R0 ≤ +∞. Then for any two positive integers k and n, RΔ(0) 0(n)(∞, f) +R Δ(k) 0(n)(0, f) ≥R δ(0) 0(n)(0, f) +R δ(0) 0(n)(a, f) +R Δ(k) 0(n)(∞, f), where a is any non zero complex number. [ABSTRACT FROM AUTHOR]

Let μ 1 be a complex number in the numerical range W (A) of a normal matrix A. In the case when no eigenvalues of A lie in the interior of W (A) , we identify the smallest convex region containing all possible complex numbers μ 2 for which μ 1 ∗ 0 μ 2 is a 2-by-2 compression of A. [ABSTRACT FROM AUTHOR]

We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type I. In our framework, each theta relation is controlled by combinatorial properties of a pair (T , P) of a regular matrix T over an imaginary quadratic field and a positive-definite Hermitian matrix P over the complex number field. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to determine all matrix representations of the real numbers. It is shown that every such representation is completely reducible, while all non-trivial irreducible representations must be of 2-dimensional and can be expressed in a unique form. It is found that those representations are essentially determined by the ways of embedding the real numbers into the complex numbers. This results in a one-to-one correspondence between the equivalent classes of irreducible representations and the equivalent classes of homomorphisms from the real number field to the complex number field. The matrix representations of the complex numbers are also determined. [ABSTRACT FROM AUTHOR]

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is (Q 1 , Q 2 , ... , Q n) , where Q n is the nth Pell–Lucas number. The formulas for the eigenvalues of such matrix are obtained. Namely, the result which can be obtained from the result of Theorem 7. (Yazlik and Taskara, J Inequal Appl 2013:394, 2013) is improved. The obtained formulas for the eigenvalues of a k-circulant matrix involving the Pell–Lucas numbers show that the result of Theorem 8. (Jing, Li and Shen, WSEAS Trans Math 12(3):341-351, 2013) (i.e. Theorem 8. (Yazlik and Taskara 2013)) is not always applicable. The Euclidean norm of such matrix is determined. The upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is (Q 1 - 1 , Q 2 - 1 , ... , Q n - 1) are also investigated. The obtained results are illustrated by examples. As a consequence of the previous results, the eigenvalues, the determinant, the Euclidean norm of a k-circulant matrix whose first row is (q 1 , q 2 , ... , q n) , where q n is the nth modified Pell number, are presented. Also, the upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is (q 1 - 1 , q 2 - 1 , ... , q n - 1) are given [ABSTRACT FROM AUTHOR]

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands. [ABSTRACT FROM AUTHOR]

We define quantum automorphisms and isomorphisms of Hadamard matrices. We show that every Hadamard matrix of size N\ge 4 has quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. These results pass also to the corresponding Hadamard graphs. We also define quantum Hadamard matrices acting on quantum spaces and bring an example thereof over matrix algebras. [ABSTRACT FROM AUTHOR]

The quaternionic Schrödinger equation ∂ ∂ t | f ⟩ = - A | f ⟩ is the crucial part of the study of quaternionic quantum mechanics and plays indispensable roles in related fields. One of the practical and special cases that has received more attention from mathematicians and physicists is that A is a Hermitian quaternion matrix. The problem can be equivalent to a Hermitian quaternion right eigenvalue problem A α = α λ by discretization. This paper, by means of a complex representation method, studies the Hermitian quaternion Schrödinger equation problem, and proposes a novel algebraic method (complex structure-preserving method) for right eigenvalue problems of Hermitian quaternion matrices. Moreover, the complex structure-preserving method is superior and formally simple compared to previous methods, and numerical experiments also demonstrate the effectiveness of the 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]

Let A be an m × n matrix with nonnegative real entries. The psd rank of A is the smallest k for which there exist two families (P 1 , ... , P m) and (Q 1 , ... , Q n) of positive semidefinite Hermitian k × k matrices such that A (i | j) = tr (P i Q j) for all i, j. Several questions on the algorithmic complexity of related matrix invariants were posed in recent literature: (i) by Stark (for the psd rank as defined above), (ii) by Goucha, Gouveia (for phaseless rank, which appears if the matrices P i and Q j are required to be of rank one in the above definition), (iii) by Gribling, de Laat, Laurent (for cpsd rank, which corresponds to the situation when A is symmetric and P i = Q i for all i). We solve these questions by proving that the decision versions of the corresponding invariants are ∃ R -complete. In addition, we give a polynomial time recognition algorithm for matrices of bounded cpsd rank. [ABSTRACT FROM AUTHOR]

The goal of this article is to present multidimensional matrix characterization of geometrically dominated double sequences. We begin this characterization with the following definition of geometrically dominated factorable double sequence space. LetG″ denote a family of double complex number sequences that are dominated by a P-convergence geometric factorable double sequence, that is, This definition will be use to present the theorems similar to the following. The four-dimensional matrix is anG″ − l″ if and only ifand Other implication will also be presented. [ABSTRACT FROM AUTHOR]

Model - based approaches to imaging, such as specialized image enhancements in astronomy, facilitate explanations of relationships between observed inputs and computed outputs. These models may be expressed with extended matrix-vector (EMV) algebra, especially when they involve only scalars, vectors, and matrices, and with n-mode or index notations, when they involve multidimensional arrays. also called numeric tensors or, simply, tensors. Although this paper features an example, inspired by exoplanet imaging, that employs tensors to reveal (inverse) 2D fast Fourier transforms in an image enhancement model, the work is actually about the tensor algebra and software, or tensor frameworks, available for model-based imaging. The paper proposes a Ricci-notation tensor (RT) framework, comprising a dual -variant index notation, with Einstein summation convention, and codesigned object-oriented software, called the RTToolbox for MATLAB. Extensions to Ricci notation offer novel representations for entrywise, pagewise, and broadcasting operations popular in EMV frameworks for imaging. Complementing the EMV algebra computable with MATLAB, the RTToolbox demonstrates programmatic and computational efficiency via careful design of numeric tensor and dual-variant index classes. Compared to its closest competitor, also a numeric tensor framework that uses index notation, the RT framework enables superior ways to model imaging problems and, thereby, to develop solutions. [ABSTRACT FROM AUTHOR]

The correlation filter object tracking algorithm has gained extensive attention from scholars in the field of tracking because of its excellent tracking performance and efficiency. However, the mathematical modeling relationships of correlation filter tracking frameworks are unclear. Therefore, many forms of correlation filters are susceptible to confusion and misuse. To solve these problems, we attempted to review various forms of the correlation filter and discussed their intrinsic connections. First, we reviewed the basic definitions of the circulant matrix, convolution, and correlation operations. Then, the relationship among the three operations was discussed. Considering this, four mathematical modeling forms of correlation filter object tracking from the literature were listed, and the equivalence of the four modeling forms was theoretically proven. Then, the fast solution of the correlation filter was discussed from the perspective of the diagonalization property of the circulant matrix and the convolution theorem. In addition, we delved into the difference between the one-dimensional and two-dimensional correlation filter responses as well as the reasons for their generation. Numerical experiments were conducted to verify the proposed perspectives. The results showed that the filters calculated based on the diagonalization property and the convolution property of the cyclic matrix were completely equivalent. The experimental code of this paper is available at https://github.com/110500617/Correlation-filter/tree/main. [ABSTRACT FROM AUTHOR]

Let a 0 , a 1 , ... , a n − 1 be real numbers and let A = C i r c (a 0 , a 1 , ... , a n − 1) be a circulant matrix with f (x) = Σ j = 0 n − 1 a j x j . First, we prove that C i r c (a 0 , a 1 , ... , a n − 1) must be invertible if the sequence a 0 , a 1 , ... , a n − 1 is a strictly monotonic sequence and a 0 + a 1 + ⋯ + a n − 1 ≠ 0. Next, we reduce the calculation of f (ε 0) f (ε) ... f (ε n − 1) for a prime n by using the techniques on finite fields, where ε is a primitive n -th root of unity. Finally, we provide two examples to explain how to use the obtained results to calculate the determinant of a circulant matrix. [ABSTRACT FROM AUTHOR]

Matrix triangular factorizations, as an intermediate step of some algorithms, are widely employed to solve scientific and engineering problems. However, there is no exact and explicit expressions on the computational complexity of triangular factorizations for general banded matrices in literatures so far. In this paper, specific and detailed descriptions on triangular factorization algorithms are presented for general banded matrices, and then by carefully dividing matrix into special blocks and with the help of the mathematical software 'Maple', exact and explicit expressions on the computational complexity of these algorithms are rigorously derived. These theoretical results are helpful for calculating the computational complexity of numerical algorithms that employ triangular factorizations, and also provide guidance for choosing appropriate algorithms for specific problems. Numerical experiments validate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euclidean operator radius. The research broadens its scope to analyze how n-tuple operators, when used as parts of 2 × 2 operator matrices, illustrate inequalities connected to the Euclidean operator radius. By using the Euclidean numerical radius and Euclidean operator norm for n-tuple operators, the study introduces a range of new inequalities. These inequalities not only set limits for the addition, multiplication, and Euclidean numerical radius of n-tuple operators but also help in establishing inequalities for the Euclidean operator radius. This process involves carefully examining the Euclidean numerical radius inequalities of 2 × 2 operator matrices with n-tuple operators. Additionally, a new inequality is derived, focusing specifically on the Euclidean operator norm of 2 × 2 operator matrices. Throughout, the research keeps circling back to the idea of finding and understanding symmetries in linear operators and matrices. The paper highlights the significance of symmetry in mathematics and its impact on various mathematical areas. [ABSTRACT FROM AUTHOR]

The B-operators (abbreviation for Brownian-type operators) are upper triangular 2\times 2 block matrix operators that satisfy certain algebraic constraints. The purpose of this paper is to characterize the weak, the strong and the uniform stability of B-operators, respectively. This is achieved by giving equivalent conditions for the convergence of powers of a B-operator in each of the corresponding topologies. A more subtle characterization is obtained for B-operators with subnormal (2,2) entry. The issue of the strong stability of the adjoint of a B-operator is also discussed. [ABSTRACT FROM AUTHOR]

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency‐resolved functional connectivity measures as a mediator using the full EEG cross‐spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive‐definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix‐valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well‐studied 50+‐year Barbados Nutrition Study dataset by comparing EEGs collected in school‐age children (5–11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains. [ABSTRACT FROM AUTHOR]

We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitary groups and the conditional probability distribution obtained by requiring the determinant of the sampled matrix be a given complex number on the complex unit circle. [ABSTRACT FROM AUTHOR]

We give real representations of $ A^n $ A n (or $ e^{t A} $ e tA ) based on $ A^nP_{\mu } $ A n P μ for a real square matrix A, where $ P_{\mu } $ P μ is the projection to the generalized eigenspace associated with an imaginary eigenvalue μ of A. Our method is based on the spectral decomposition theorem. As applications, we can easily obtain realifications of representations of solutions of inhomogeneous linear difference equations with constant coefficients. [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]

We investigate selfadjointness in sense of Hilbert space rigging and related questions. We proved that this generalized selfadjointness of some operator, which acts from positive into negative space, is equivalent to ordinary selfadjointness of some modification of this operator in basic (“zero”) space. Also we consider operators generated by classical and generalized Jacobi Hermitian matrices, their selfadjointness and generalized selfadjointness in sense of weight Hilbert space rigging. Some sufficient conditions of generalized selfadjointness of these operators are proved. Using obtained results we explaine possibility of construction of example of gereralized selfadjoint opearator which is not selfadjoint in classical sence. [ABSTRACT FROM AUTHOR]

. We consider a minimal realization of a rational matrix. We perturb all the coefficients of matrix polynomial and some coefficients from the realization part present in the realization form of rational matrix. We derive explicit computable formulae for backward error of approximate eigenvalues and eigenpairs of regular rational matrix. We also determine minimal perturbations for all the coefficients of matrix polynomial and some coefficients from the realization part for which approximate eigenvalues are exact eigenvalues of the perturbed rational matrix. [ABSTRACT FROM AUTHOR]

A novel unconstrained optimization model named weighted trace-penalty minimization (WTPM) is proposed to address the extreme eigenvalue problem arising from the full configuration interaction (FCI) method. Theoretical analysis shows that the global minimizers of the WTPM objective function are the desired eigenvectors, rather than the eigenspace. Analyzing the condition number of the Hessian operator in detail contributes to the determination of a near-optimal weight matrix. With the sparse feature of FCI matrices in mind, the coordinate descent (CD) method is adapted to WTPM and results in the WTPM-CD method. The reduction of computational and storage costs in each iteration shows the efficiency of the proposed algorithm. Finally, the numerical experiments demonstrate the capability to address large-scale FCI matrices. [ABSTRACT FROM AUTHOR]

We investigate the problem of scaling a given positive definite matrix A to achieve a prescribed eigenpair (λ,v), by way of a diagonal scaling D*AD. We consider the case where D is required to be positive, as well as the case where D is allowed to be complex. We generalize a few classical results, and then provide a partial answer to a question of Pereira and Boneng regarding the number of complex scalings of a given 3×3 positive definite matrix A. [ABSTRACT FROM AUTHOR]

In order to further connect structured matrices and integer sequences, r -circulant matrices involving the generalized Narayana numbers are considered. Estimates for spectral norms bounds of such matrices are presented and their eigenvalues are determined. Moreover, the conditions under which the circulant matrix and the skew circulant matrix involving generalized Narayana numbers are invertible are given. In particular, it is shown that every circulant matrix with Narayana numbers is necessarily invertible. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mixed solution of reduced biquaternion matrix equation Σni=1 AiXiBi=Ewith sub-matrix constraints. With the help of LC-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion matrices, the reduced biquaternion matrix equation (1.1) can be transformed into linear equations. A systematic method, GH-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the least squares problem of linear equations. Meanwhile, we give the necessary and suficient conditions for the compatibility of reduced biquaternion matrix equation (1.1) under sub-matrix constraints. Numerical examples are given to demonstrate the results. The method proposed in this paper is applied to color image restoration. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of general coupled linear matrix equations over the complex number field. The mentioned coupled linear matrix equations contain the unknown complex matrix groups $$X=(X_1,X_2,\ldots ,X_q)$$ and $$Z=(Z_1,Z_2,\ldots ,Z_q)$$ . The conjugate and transpose of the unknown matrices $$X_i$$ and $$Z_i$$ , $$i\in I[1,q]$$ , appear in the considered coupled linear matrix equations. An iterative algorithm is presented to determine the unknown matrix groups $$X$$ and $$Z$$ such that $$X$$ and $$Z$$ are the groups of the generalized reflexive and anti-reflexive matrices, respectively. The proposed algorithm determines the solvability of the general coupled linear matrix equations over the generalized reflexive and anti-reflexive matrices, automatically. When the general coupled linear matrix equations are consistent over the generalized reflexive and anti-reflexive matrices, it is shown that the algorithm converges within finite number of steps, in the exact arithmetic. In addition, the optimal approximately generalized reflexive and anti-reflexive solution groups to the given arbitrary matrix groups $$\Gamma _x=(\Gamma _{1x},\Gamma _{2x},\ldots ,\Gamma _{qx})$$ and $$\Gamma _z=(\Gamma _{1z},\Gamma _{2z},\ldots ,\Gamma _{qz})$$ are derived. Finally, some numerical results are given to illustrate the validity of the presented theoretical results and feasibly of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Let Tn be the set of complex Toeplitz n × n matrices. The paper describes the matrices U in the linear group GLn(ℂ) such that ∀ A ∈ T n → U − 1 AU ∈ T n and also the matrices U ∈ GLn(ℂ) such that ∀ A ∈ T n → U − 1 A U ¯ ∈ T n. [ABSTRACT FROM AUTHOR]

To improve the accuracy of color image completion with missing entries, we present a recovery method based on generalized higher-order scalars. We extend the traditional second-order matrix model to a more comprehensive higher-order matrix equivalent, called the "t-matrix" model, which incorporates a pixel neighborhood expansion strategy to characterize the local pixel constraints. This "t-matrix" model is then used to extend some commonly used matrix and tensor completion algorithms to their higher-order versions. We perform extensive experiments on various algorithms using simulated data and publicly available images. The results show that our generalized matrix completion model and the corresponding algorithm compare favorably with their lower-order tensor and conventional matrix counterparts. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide new representations and computations of the generalized Moore-Penrose inverse. Based on the Moore-Penrose inverse, group inverse, Bott-Duffin inverse and certain projections, some representations for the generalized Moore-Penrose inverse are given. An equivalent condition for the continuity of the generalized Moore-Penrose inverse is proposed. Splitting methods and successive matrix squaring algorithm for computing the generalized Moore-Penrose inverse are presented. [ABSTRACT FROM AUTHOR]

Minimum-phase properties are well-understood for scalar functions where they can be used as physical constraint for phase reconstruction. Existing scalar applications of the latter in geophysics include, for example the reconstruction of transmission from acoustic reflection data, or multiple elimination via the augmented acoustic Marchenko method. We review scalar minimum-phase reconstruction via the conventional Kolmogorov relation, as well as a less-known factorization method. Motivated to solve practice-relevant problems beyond the scalar case, we investigate (1) the properties and (2) the reconstruction of minimum-phase matrix functions. We consider a simple but non-trivial case of 2 × 2 matrix response functions associated with elastodynamic wavefields. Compared to the scalar acoustic case, matrix functions possess additional freedoms. Nonetheless, the minimum-phase property is still defined via a scalar function, that is a matrix possesses a minimum-phase property if its determinant does. We review and modify a matrix factorization method such that it can accurately reconstruct a 2 × 2 minimum-phase matrix function related to the elastodynamic Marchenko method. However, the reconstruction is limited to cases with sufficiently small differences between P - and S -wave traveltimes, which we illustrate with a synthetic example. Moreover, we show that the minimum-phase reconstruction method by factorization shares similarities with the Marchenko method in terms of the algorithm and its limitations. Our results reveal so-far unexplored matrix properties of geophysical responses that open the door towards novel data processing tools. Last but not least, it appears that minimum-phase matrix functions possess additional, still-hidden properties that remain to be exploited, for example for phase reconstruction. [ABSTRACT FROM AUTHOR]

Let A be an unital algebra over the complex field C . A linear map ϕ from A into itself is called a Lie centralizer at a given point G ∈ A if ϕ ([ S , T ]) = [ S , ϕ (T) ] = [ ϕ (S) , T ] for all S , T ∈ A with ST = G. The aim of this paper is to give a description of Lie centralizers at an arbitrary but fixed point on triangular algebras. These results are then applied to nest algebras and upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

A novel approach that does not require the number of sources as a priori is proposed to estimate the direction of arrival (DOA) based on a sparse non-uniform linear antenna array. To ensure the identifiability of the DOA, a specific configuration scheme of sparse array is designed. Based on this specific sparse array, firstly the fourth-order cumulant (FOC) is adopted to eliminate the impact imposed by Gaussian noise. Secondly, to circumvent eigenvalue decomposition or singular value decomposition, a propagator is constructed by using a Hermitian FOC matrix and a hyperparameter. Finally, a projection onto an irregular Toeplitz set is proposed to further improve estimation accuracy. [ABSTRACT FROM AUTHOR]

The Neumann–Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation. They can be used as input data for the construction of a number of quantum invariants that include the loop invariants, the 3D-index and state-integrals. We define a twisted version of Neumann–Zagier matrices, describe their symplectic properties, and show how to compute them from the combinatorics of an ideal triangulation. As a sample application, we use them to define a twisted version of the 1-loop invariant (a topological invariant) which determines the 1-loop invariant of the cyclic covers of a hyperbolic knot complement, and conjecturally is equal to the adjoint twisted Alexander polynomial. [ABSTRACT FROM AUTHOR]

In simulating Quantum Computing by using the circuit model the size of the matrices to deal with, together with the number of products and additions required to apply every quantum gate becomes a really hard computational restriction. This paper presents a data structure, called Functional Matrices, which is the most representative feature of QSimov quantum computing simulator which is also provided and tested. A comparative study of the performance of Functional Matrices with respect to the other two most commonly used matrix data structures, dense and sparse ones, is also performed and summarized within this work. [ABSTRACT FROM AUTHOR]

We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices. [ABSTRACT FROM AUTHOR]

The purpose of this note is to present upper bounds estimations for the numerical radius of a products and Hadamard products of special matrices, including sectorial and accretivedissipative matrices. [ABSTRACT FROM AUTHOR]

In this paper, we define the bi-periodic hyper-Horadam sequence {w(k)n }n∈N and present its combinatorial properties. Moreover, we obtain upper and lower bounds for the spectral norms of different forms of the r -circulant and geometric circulant matrices with the bi-periodic hyper-Horadam sequence. Then we give some bounds for the spectral norms of the Kronecker and Hadamard products of these matrices. [ABSTRACT FROM AUTHOR]

Several inequalities are presented for sector matrices. First, an analogue of the AM-GM inequality is established. As applications of this inequality, similar inequalities are presented for singular values and norms. Finally, some unitarily invariant norm inequalities are obtained for sector matrices. [ABSTRACT FROM AUTHOR]

In this paper we establish some necessary and sufficient solvability conditions for a system of quaternary coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. An expression of the general solution to this system is given when it is solvable. Also, a numerical example is presented to illustrate the main result of this paper. The findings of this paper widely generalize the known results in the literature. The main results are also valid over the real number field and the complex number field. [ABSTRACT FROM AUTHOR]