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Let $ \mathcal {O} $ O denote the ring of integers of a quadratic field $ \mathbb {Q}(\sqrt {-7}) $ Q (− 7). In 2022, Murtuza and Garge [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022.] gave a necessary and sufficient condition for a diagonal quadratic form $ a_1X_1^2+a_2X_2^2+a_3X_3^2 $ a 1 X 1 2 + a 2 X 2 2 + a 3 X 3 2 where $ a_i\in \mathbb {\mathcal {O}} $ a i ∈ O for $ 1\leq i \leq ~3 $ 1 ≤ i ≤ 3 for representing all $ 2\times 2 $ 2 × 2 matrices over $ \mathcal {O} $ O . Let K denote a quadratic field such that its ring of integers $ \mathcal {O}_K $ O K is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that $ \mathbb {Q}(\sqrt {-7}) $ Q (− 7) is the only imaginary quadratic field with the above properties. Let $ D_K $ D K denote the discriminant of K. We have $ D_K\equiv 1(\text{mod }8) $ D K ≡ 1 (mod 8) if and only if 2 is a product of two distinct primes in $ \mathcal {O}_K $ O K . With $ \mathcal {O}_K $ O K as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form $ {\sum _{i=1}^{m}a_iX_i^2} $ ∑ i = 1 m a i X i 2 where $ a_i\in \mathcal {O}_K $ a i ∈ O K , $ 1\leq i \leq m $ 1 ≤ i ≤ m to represent all $ 2\times 2 $ 2 × 2 matrices over $ \mathcal {O}_K $ O K . This result is a conjecture stated in [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022]. [ABSTRACT FROM AUTHOR]

A quasi-Toeplitz M-matrix A is an infinite M-matrix that can be written as the sum of a semi-infinite Toeplitz matrix and a correction matrix. This paper is concerned with computing the square root of invertible quasi-Toeplitz M-matrices which preserves the quasi-Toeplitz structure. We show that the Toeplitz part of the square root can be easily computed through evaluation/interpolation. This advantage allows us to propose algorithms solely for the computation of correction part, whence we propose a fixed-point iteration and a structure-preserving doubling algorithm. Additionally, we show that the correction part can be approximated by solving a nonlinear matrix equation with coefficients of finite size followed by extending the solution to infinity. Numerical experiments showing the efficiency of the proposed algorithms are performed. [ABSTRACT FROM AUTHOR]

In this paper, we study the Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems A x = b and obtain the structure-preserving algorithms of Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems A x = b . The convergence and computational cost of these iteration methods are discussed. Numerical examples are given to demonstrate the efficiency of structure-preserving algorithms of Gauss-Seidel iteration and successive over-relaxation iteration methods. As an application, we apply two kinds of structure-preserving iterative algorithms to solve elliptic biquaternion linear systems A x = b . [ABSTRACT FROM AUTHOR]

In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and error matrix decrease monotonically. Numerical tests are presented to show the efficiency of the proposed method and confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the following three classes of constrained matrix inequalities are considered: G ∗ X + X ∗ G ≥ H (> H) , s.t. A X B = D ; G ∗ X + X ∗ G ≥ H (> H) , s.t. A X = C , X B = D ; G ∗ X D + D ∗ X ∗ G ≥ H (> H) , s.t. A X = B. The solvability conditions and the explicit representations for the general solutions are provided when the stated conditions hold. [ABSTRACT FROM AUTHOR]

We propose an effective matrix iteration method and a novel zeroing neural network (NZNN) model for finding numerical commuting solutions of the (time-invariant) Yang–Baxter-like matrix equation in this paper. The proposed matrix iteration method has a second-order convergence speed, and it is proved to be stable. How to proceed with initial value selection and termination criteria are discussed. Meanwhile, two numerical experiments are adopted to illustrate the superiority of the proposed matrix iteration method in computational efficiency. The NZNN model based on Tikhonov regularization for solving the time-invariant Yang–Baxter-like matrix equation is given. Besides, numerical results are provided to substantiate the efficiency, availability and superiority of the developed NZNN model for time-invariant Yang–Baxter-like matrix equation problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an encryption method for color image data transmission based on Sylvester-type quaternion matrix equations. We first study the solvability conditions and general solution to a system of Sylvester-type quaternion matrix equations. We derive some practical necessary and sufficient conditions for the existence of a solution to this system by using a simultaneous decomposition of quaternion matrices. We present the expression of the general solution to the system when the solvability conditions are satisfied. Based on the form of this system, we consider the applications of this system in color image data transmission. Moreover, we provide some algorithms and examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability conditions of some Sylvester-type quaternion matrix equations. We establish some practical necessary and sufficient conditions for the existence of solutions of a Sylvester-type quaternion matrix equation with five unknowns through the corresponding equivalence relations of the block matrices. Moreover, we present some solvability conditions to some Sylvester-type quaternion matrix equations, including those involving Hermicity. The findings of this article extend related known results. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is, in the first place, to endow the space of quiddity sequences corresponding to the usual frieze patterns with a SET operad structure, and second to introduce and study M -quiddity sequences where M is a monodromy block matrix of order two. Also, we examine related topics such as the possibility of defining matrix-valued friezes patterns and noncommutative signed Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we initially introduce several theoretical results regarding the existence of solutions for the nonlinear tensor equation X + A T ∗ N X - 1 ∗ N A = I. Then, we present iterative algorithms without inversion to determine solutions of the nonlinear tensor equation. We demonstrate the convergence of these algorithms towards solving the nonlinear tensor equation. Finally, we provide various numerical experiments to illustrate their effectiveness and accuracy. [ABSTRACT FROM AUTHOR]

The automated removal of layers by scarfing to repair carbon fibre reinforced plastics (CFRP) has been investigated with vacuum suction blasting (VSB) showing promising results. However, its removal mechanism and particle behaviour were not yet fully understood. The removal leads to cavities on the surface and the central area is less removed after longer static blasting times. Further theoretical considerations and experiments of measured forces and velocities of the VSB particles can explain these effects to optimize the large-scale removal. Former studies on abrasive blasting used simplified approaches with incompressible Bernoulli equations taking the same velocities of particles and carrier medium. For VSB, the air is much faster than the particles and more complex equations are derived in this study. Additionally, it is taken into account, that the impinging forces act against the vacuum forces. To specify the power and removal mechanism of VSB for large scale removal, forces and velocities of the blasting particles are studied. Calculations by compressible Bernoulli equations, measurements of blasting particle velocities and impact forces give a good approximation. The air velocities and forces are calculated and compared with the measurements. Forces are measured by a piezo-electric sensor. Particle velocities are measured with a Particle Image Velocimetry (PIV) system tracking the abrasive particle movement through a transparent outer nozzle. The influence of different blasting parameters on the forces and velocities are identified. Results correspond to the material removal from former experiments and indicate the favourable process parameter set. Despite the impinging forces are in general low, they lead together with the particle velocity to the large-scale removal rate. The measured particle directions explain the removal behaviour of this VSB process. The measured velocities match to those well-known from grinding with bonded grinding tools. Article highlights: Force and velocity measurements of vacuum suction blasting process PIV measurements of blasting particles Forces calculated and measured by piezo-electric sensor. [ABSTRACT FROM AUTHOR]

We seek the solution of the coupled continuous-time algebraic Riccati equations, arising in the optimal control of Markovian jump linear systems. Newton's method is applied to construct the solution, under a mild and natural stabilizability assumption, leading to some coupled Lyapunov equations. Iterative methods of O (n 3) computational complexity for the coupled Lyapunov equations and the corresponding Newton's methods for the coupled continuous-time algebraic Riccati equations are analyzed. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

A square matrix A over the field of complex numbers is said to be Hermitian if A = A ∗ , the conjugate transpose of A, while Hermitian matrices are known to be an important class of matrices. In addition to the definition, a Hermitian matrix can be characterized by some other matrix equalities. This fact can be described in the implication form f (A , A ∗) = 0 ⇔ A = A ∗ , where f (·) denotes certain ordinary algebraic operation of A and A ∗ . In this note, we show two special cases of the equivalent facts: A A ∗ A = A ∗ A A ∗ ⇔ A 3 = A A ∗ A ⇔ A = A ∗ without assuming the invertibility of A through the skillful use of decompositions and determinants of matrices. Several consequences and extensions are presented to a selection of matrix equalities composed of multiple products of A and A ∗ . [ABSTRACT FROM AUTHOR]