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Algebraic Riccati equations (AREs) have been extensively applied in linear optimal control problems and many efficient numerical methods were developed. The stabilizing (or almost stabilizing) solution that all eigenvalues of its closed-loop matrix are contained in the open (or closed) unit disk of the complex plane has attracted the most attention among all Hermitian solutions of the ARE in the past works. Nevertheless, it is an interesting and challenging issue in finding the extremal solutions of AREs which play an important role in the applications. The contribution of this paper is twofold. Firstly, the existence of these extremal solutions is established under the framework of fixed-point iteration. Secondly, an accelerated fixed-point iteration (AFPI) based on the semigroup property is developed for computing four extremal solutions of the discrete-time algebraic Riccati equation, which has not appeared in the existing literature. In addition, we prove that the convergence of the AFPI is at least R-suplinear with order r > 1 under some mild assumptions. Numerical examples are shown to illustrate the feasibility and accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the interval coupled Sylvester matrix equations which include the well-known (generalized) Sylvester and Lyapunov matrix equations in both real and interval forms. Assuming that certain matrices are simultaneously diagonalizable, we present a fast and efficient approach for enclosing the solution set of the interval coupled Sylvester matrix equations. Our approach, which is a modification of the Krawczyk operator, enables us to reduce the computational complexity considerably. Some numerical tests are given to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

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]

In this paper, we consider skew-Hermitian solution of coupled generalized Sylvester matrix equations encompassing ∗ -hermicity over complex field. The compact formula of the general solution of this system is presented in terms of generalized inverses when some necessary and sufficient conditions are fulfilled. An algorithm and a numerical example are provided to validate our findings. A numerical example is carried out using determinantal representations of the Moore–Penrose inverse. [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]

Linear coupled matrix equations are widely utilized in applications, including stability analysis of control systems and robust control. In this paper, we establish the necessary and sufficient conditions for the consistency of the system of linear coupled matrix equations and derive an expression of the corresponding general solution (where it is solvable) over quaternion. Additionally, we investigate the necessary and sufficient conditions for the system of linear coupled matrix equations with construct to have a solution and derive a formula of its general solution (where it is solvable). Finally, an algorithm and an example were provided in order to further illustrate the primary outcomes of this paper. [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 establish some necessary and sufficient conditions for the solvability to a system of two-sided Sylvester-like tensor equations over the quaternion algebra. We also construct an expression of the general solution to the system when it is solvable. As an application, we give some solvability conditions and expressions of the η -Hermitian solutions to some systems of two-sided Sylvester-like quaternion tensor equations. We also provide some algorithms and numerical examples to illustrate the main results of this paper. [ABSTRACT FROM AUTHOR]

This paper presents several new expressions for the W-weighted Drazin and core-EP inverse of a matrix based on Urquhart formula. These expressions naturally extend the classic results for the Drazin and core-EP inverse and some of them improve the existing results in the literature. This paper also presents new finite methods for computing the W-weighted Drazin and core-EP inverse through the sequential procedure of full rank factorizations of Cline. [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]

Purpose: In this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory. Design/methodology/approach: The authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations. Findings: The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices. Originality/value: Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$. [ABSTRACT FROM AUTHOR]

In this paper, based on semi-tensor product of quaternion matrices, we explore the operation conclusion of quaternion matrices under vector representation, which can realize the transformation from quaternion matrix equation to quaternion vector matrix equation. Then using complex representation of quaternion matrix and Moore–Penrose generalized inverse, we obtain the expression of the minimal norm least squares (anti)-Hermitian solution of the system of quaternion generalized Sylvester matrix Eq. (1.1). We compare the algorithm in this paper with the other algorithms, and the corresponding numerical experiments reflect the feasibility and advantage of the algorithm in this paper. At the same time, we apply the algorithm in this paper to the symmetric color digital image restoration model. [ABSTRACT FROM AUTHOR]

Sylvester-like matrix equations are encountered in many areas of control engineering and applied mathematics. In this paper, we construct some necessary and sufficient conditions for the system of Hermitian mixed type generalized Sylvester matrix equations to have a solution. The closed form formula to compute the general solution is also established when solvability conditions are satisfied. An algorithm and a numerical example are provided to validate our findings. [ABSTRACT FROM AUTHOR]

This paper studies a system of three Sylvester-type quaternion matrix equations with ten variables A i X i + Y i B i + C i Z i D i + F i Z i + 1 G i = E i , i = 1 , 3 ¯ . We derive some necessary and sufficient conditions for the existence of a solution to this system in terms of ranks and Moore-Penrose inverses of the matrices involved. We present the general solution to the system when the solvability conditions are satisfied. As applications of this system, we provide some solvability conditions and general solutions to some systems of quaternion matrix equations involving ϕ-Hermicity. Moreover, we give some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature. [ABSTRACT FROM AUTHOR]

A square complex matrix A is EP if it satisfies R (A) = R (A ∗) (or, equivalently, R (A) ⊆ R (A ∗)). This paper aims to introduce two natural generalizations of EP matrices: a square matrix A is defined to be left or right power-EP if it satisfies R (A m) ⊆ R (A ∗) or N (A ∗) ⊆ N (A m) for some integer m ≥ 0 , respectively. Some properties, characterizations and applications of these two classes of matrices are given. [ABSTRACT FROM AUTHOR]

Given a bilinear form on C n , represented by a matrix A ∈ C n × n , the problem of finding the largest dimension of a subspace of C n such that the restriction of A to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix B such that the equation X ⊤ A X = B is consistent (here X ⊤ denotes the transpose of the matrix X). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation X ⊤ A X = B to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices A ∈ C n × n . To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix A, which is a direct sum of blocks of three types. The condition is valid for all matrices A except those whose CFC contains blocks, of one of the types, with size smaller than 3. However, we show that the condition is necessary for all matrices A. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in computing the minimal positive solution of an algebraic Riccati equation arising in transport theory. The coefficient matrices of this equation have two parameters α and c. When 0 < α < 1 , 0 < c < 1 , the existing numerical algorithms can solve its minimal positive solution very quickly and efficiently. However, these algorithms do not converge or converge slowly to the minimal positive solution for α = 0 and c = 1 . In this case, we propose two methods to improve the effectiveness of these algorithms. The first method is a modified double shift technique to transform the original algebraic Riccati equation into a new one, and the two equations have the same minimal positive solution. The second method is a deflating technique to transform the original equation into a new low-order algebraic Riccati equation, and we give the relationship between the minimal positive solutions of these two equations. The minimal positive solutions of two new equations both can be effectively solved by the existing algorithms. We show that two new equations both maintain the symmetry. Moreover, two new equations are M-matrix algebraic Riccati equations. Numerical experiments are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Sylvester-type matrix equations have wide applications in system science, control theory, and so on. In this paper, we consider a constrained system of Sylvester-type matrix equations over the quaternions. We first derive the necessary and sufficient conditions for the system to have a solution and propose a formula of its general solution when it is solvable. As an application, we then investigate the η -Hermitian solution of a system. Moreover, we also give a numerical example to illustrate the main findings of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider a linear constraint problem of Hermitian unitary symplectic matrices and its approximation. By constructing a simple unitary matrix U, we verify that Hermitian unitary symplectic matrices are unitary similar to block diagonal Hermitian unitary matrices via U, which simplifies and is crucial to solving the linear constraint problem, and is a special feature of this paper. Then, we solve the linear constraint problem completely, that is deriving the sufficient and necessary conditions of it and inducing Hermitian unitary symplectic solutions to it. We also obtain its optimal approximate solutions. Furthermore, the Procrustes problem of Hermitian unitary symplectic matrices is considered when the linear constraint problem has no solution. [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, 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]

In this paper, we establish some necessary and sufficient conditions for the solvability to a system of five quaternion matrix equations in terms of the Moore–Penrose inverse and the rank of a matrix, and give an expression of the general solution to the system when it is consistent. As an application, we investigate an η -Hermicity solution of a system. Moreover, we present a numerical example to illustrate the main results of this paper. [ABSTRACT FROM AUTHOR]

This paper focuses on finding solutions of generalized Sylvester-type matrix equations over the quaternion skew-field. We express the general least–squares solutions, and perhermitian, skew-perhermitian least-squares solutions of A X B + C Y D = E and A X B + C X D = E over the quaternion skew-field in terms of a vec operator (defined specifically for matrices over the quaternion skew-field) and the Moore–Penrose pseudoinverse. In addition, characterizations that facilitate the computation of the least-squares solutions closest to prescribed quaternion matrices are deduced. We illustrate our theoretical findings on several numerical examples, most of which originate from color image restoration via Tikhonov regularization. [ABSTRACT FROM AUTHOR]

This paper mainly gives some sufficient and necessary conditions for an element in a ring to be EP , partial isometry and strongly EP by some equalities, using the solutions of certain equations and constructing the invertible elements in a ring. [ABSTRACT FROM AUTHOR]

In this paper, we derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the found form and fast Fourier transform, we propose a new algorithm for solving both discrete-time and continuous-time large-scale algebraic Riccati equations with low-rank structure. It works without unnecessary assumptions, complicated shift selection strategies, or matrix calculations of the cubic order with respect to the problem scale. Numerical examples are given to illustrate its features. Besides, we show that it is theoretically equivalent to several algorithms existing in the literature in the sense that they all produce the same sequence under the same parameter setting. [ABSTRACT FROM AUTHOR]

In this paper, we establish a different approach for solving the system of three coupled two-sided Sylvester-type quaternion matrix equations A i X i B i + C i X i + 1 D i = E i , i = 1 , 3 ¯ . We give some new necessary and sufficient conditions for the existence of a solution to this system in terms of Moore-Penrose inverses of the matrices involved. We show that these new solvability conditions are equivalent with the solvability conditions which were presented in a recent paper [Linear Algebra Appl. 2016;496:549–593]. The general solution to the system is given when the solvability conditions are satisfied. Applications that are discussed include the solvability conditions and general η-Hermitian solution to a system of quaternion matrix equations. [ABSTRACT FROM AUTHOR]

In this paper we consider linear gyroscopic mechanical systems.More precisely, we consider the perturbation theory for stable gyroscopic systems. We present new relative perturbation bounds for the eigenvalues as well as the bounds for the perturbation of the corresponding eigenspaces. Derived bounds are dependent only on system matrices of the original and perturbed systems. The quality of obtained results is illustrated in the numerical experiment. [ABSTRACT FROM AUTHOR]

In this paper, we establish some efficient iteration methods for solving the general coupled discrete-time periodic matrix equation. More concretely, we propose some structure preserving matrix versions of subspace methods, such as bi-conjugate gradient, bi-conjugate residual, conjugate gradient squared, bi-conjugate gradient stabilized, generalized minimal residual and restarted generalized minimal residual methods. We demonstrate experimentally that the proposed iteration methods are feasible and efficient for the periodic matrix equation. [ABSTRACT FROM AUTHOR]