Bisymmetric Matrices Research Articles

Inverse eigenvalue problem of bisymmetric nonnegative matrices

This paper considers an inverse eigenvalue problem for‎ bisymmetric nonnegative matrices‎. ‎We first discuss the specified‎ structure of the bisymmetric matrices‎. ‎Then for a given set of real‎ numbers of order maximum ‎‎five‎ with special conditions‎, ‎we construct‎ a nonnegative bisymmetric matrix such that the given set is its spectrum‎‎. ‎Finally, we solve the problem for ‎arbitrary ‎order $n$ in the special case of the spectrum‎.

Least‐squares partially bisymmetric solutions of coupled Sylvester matrix equations accompanied by a prescribed submatrix constraint

The coupled Sylvester matrix equations (CSMEs) appear frequently in various fields of mathematics and engineering such as in control systems and signal processing. In this investigation, we establish and analyze the generalized conjugate directions (GCDs) method for solving the CSMEs over partially bisymmetric matrices X and Y with a prescribed submatrix constraint. We show that the GCD method with the arbitrary initial bisymmetric matrices can compute the least‐squares partially bisymmetric solutions with a prescribed submatrix constraint within a finite number of iterations in the absence of round‐off errors. Numerical examples illustrate the efficiency and simplicity of the GCD method and confirm the theoretical results.

Modeling inherited physiological structures based on hyperbolic numbers

The article is devoted to using the mathematics of multi-dimensional hyperbolic numbers and their matrix representations for modeling of different inherited biosystems, which are parts of the holistic body. The list of considered models includes phyllotaxis sequences; Punnet squares of the Mendelian genetics; the main psychophysical Weber-Fechner law, and some others. This modeling approach reveals hidden structural interconnections among different biosystems and leads a deeper understanding of their commonality related to the commonality of their genetic basis. At the same time, it shows the structural connection of inherited biosystems with formalisms of the theory of resonances of oscillatory systems with many degrees of freedom. This complements works of many authors - from antiquity to the present day - on the importance of coordinated oscillations and resonances in the life of living organisms. The modeling approach based on hyperbolic numbers and their bisymmetric matrices reveals that the structural commonality of different inherited biosystems is related to the harmonic progression 1/n and the harmonic mean, which are long known in arts and culture. They are used in the Pythagorean teaching on the musical harmony and the aesthetics of proportions. Besides, the using - for biosystems modeling - of multi-dimensional hypercomplex numbers, which are one of the main tools in modern mathematical natural sciences, opens a bridge between these sciences and biology for their mutual enrichment.

Self-dual codes using bisymmetric matrices and group rings

In this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F4+uF4 together with the well known extension and neighbour methods, we construct new self-dual codes of length 68. In particular, we find 41 new codes of length 68 that were not known in the literature before.

Bisymmetric matrices and symmetrologic researches in genetic biomechanics

Bisymmetric matrices and symmetrologic researches in genetic biomechanics

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

AbstractThe inverse eigenvalue problem appears repeatedly in a variety of applications. The aim of this paper is to study a quadratic inverse eigenvalue problem of the form AXΛ2 + BXΛ + CX = 0 where A, B and C should be partially bisymmetric under a prescribed submatrix constraint. We derive an efficient matrix method based on the Hestenes‐Stiefel (HS) version of biconjugate residual (BCR) algorithm for solving this constrained quadratic inverse eigenvalue problem. The theoretical results demonstrate that the matrix method solves the constrained quadratic inverse eigenvalue problem within a finite number of iterations in the absence of round‐off errors. Finally we validate the accuracy and efficiency of the matrix method through the numerical results.

Least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints

The inverse eigenvalue problems play an important role in broad application areas such as system identification, Hopfield neural networks, control design, mass–spring system and molecular spectroscopy. This paper proposes an algorithm that yields a new method to efficiently and accurately compute the partially bisymmetric solutions (M,C,K) under prescribed submatrix constraints of the quadratic inverse eigenvalue problem MXΛ2+CXΛ+KX=0. The algorithm is developed based on the conjugate gradient normal equations residual minimizing (CGNR) method. We discuss the convergence properties of the algorithm. Finally, the performance of the algorithm is tested on two numerical examples and compared to the previous algorithm.

The inverse of bisymmetric matrices

An matrix is called a bisymmetric matrix if and . In this paper, we show that the inverse of bisymmetric matrix can be represented by recurrence formulas. First, we obtain an explicit inverse formula of bisymmetric matrices. Second, a recurrence algorithm for the inverse is given. Finally, an example is provided to verify the feasibility of the algorithms.

Term rank preservers of bisymmetric matrices over semirings

In this article, we introduce another standard form of linear preservers. This new standard form provides characterizations of the linear transformations on the set of bisymmetric matrices with zer...

Iterative solutions of generalized inverse eigenvalue problem for partially bisymmetric matrices

In this paper, for given , , an iterative algorithm is constructed to find the solutions of generalized inverse eigenvalue problem , where A and B should be partially bisymmetric under a prescribed submatrix constraint. For any initial constrained matrices, a solution pair can be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The least norm solution can be obtained by choosing a special kind of initial matrix pencil. In addition, the unique optimal approximation solution to a given matrix pencil in the solution set of the above problem can also be obtained by this iterative algorithm. Numerical examples are given to illustrate efficiency of the proposed algorithm.

Solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices

In many areas of principal component analysis, biology, electricity, solid mechanics, automatics control theory and vibration theory, linear matrix equations can be encountered. The presented paper proposes first an iterative method for finding the generalized bisymmetric solution to the generalized coupled Sylvester matrix equations. Second, when the generalized coupled Sylvester matrix equations matrix are consistent, for any generalized bisymmetric initial iterative matrix pair, we can obtain the generalized bisymmetric solution within finite iterative steps in the absence of round-off errors. Furthermore, the optimal approximate generalized bisymmetric solution of the matrix equation for a given generalized bisymmetric matrix can be obtained by finding the least-norm generalized bisymmetric solution of new generalized coupled Sylvester matrix equations. Finally, numerical examples are presented to support the theoretical results of this paper.

Persymmetric and bisymmetric nonnegative inverse eigenvalue problem

The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n×n entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for n≥5. If the matrix A is required to be persymmetric (bisymmetric), the problem will be called persymmetric (bisymmetric) nonnegative inverse eigenvalue problem (PNIEP) (BNIEP). Persymmetric and bisymmetric matrices are common in physical sciences and engineering. They arise, for instance, in the control of mechanical and electric vibrations. A persymmetric version of a perturbation result, due to Rado and presented by H. Perfect in [5], is developed and used to give sufficient conditions for the PNIEP to have a solution. Our results generate an algorithmic procedure to compute the solution matrix.

Solving the system of linear operator equations over generalized bisymmetric matrices

This paper proposes an iterative method based on the conjugate gradient method on the normal equations for finding the generalized bisymmetric solution [Formula: see text] to the system of linear operator equations [Formula: see text] where [Formula: see text] are linear operators. By the iterative method, the solvability of this system over the generalized bisymmetric matrix [Formula: see text] can be determined automatically. When the system of linear operator equations is consistent over the generalized bisymmetric matrix [Formula: see text], the iterative method with any generalized bisymmetric initial iterative matrix [Formula: see text] can compute the generalized bisymmetric solution within a finite number of iterations in the absence of roundoff errors. In addition, by the proposed iterative method, the least Frobenius norm generalized bisymmetric solution can be derived when a special initial generalized bisymmetric matrix is chosen. Finally, two numerical examples are presented to support the theoretical results of this paper.

Linear parameterized inverse eigenvalue problem of bisymmetric matrices

In this paper, we consider the linear parameterized inverse eigenvalue problem of bisymmetric matrices which is described as follows:Problem IGiven real bisymmetric matrices B0,B1,B2,…,Bn, and real numbers λ1⁎,λ2⁎,…,λn⁎, find values of c=(c1,c2,…,cn)T such that the eigenvalues of the matrixB(c)=B0+c1B1+c2B2+⋯+cnBn are precisely λ1⁎,λ2⁎,…,λn⁎.We first establish the structure of bisymmetric matrices. Based on the Newton-type iterative method, we then turn Problem I into the inverse eigenvalue problems of symmetric matrices with small sizes, and solve these problems. Finally, we present numerical examples to illustrate our results.

Solving coupled matrix equations over generalized bisymmetric matrices

In this paper, an iterative algorithm is established for finding the generalized bisymmetric solution group to the coupled matrix equations (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases). It is proved that proposed algorithm consistently converges to the generalized bisymmetric solution group for any initial generalized bisymmetric matrix group. Finally a numerical example indicates that proposed algorithm works quite effectively in practice.

The generalised Sylvester matrix equations over the generalised bisymmetric and skew-symmetric matrices

A matrix P is called a symmetric orthogonal if P = P T = P −1. A matrix X is said to be a generalised bisymmetric with respect to P if X = X T = PXP. It is obvious that any symmetric matrix is also a generalised bisymmetric matrix with respect to I (identity matrix). By extending the idea of the Jacobi and the Gauss–Seidel iterations, this article proposes two new iterative methods, respectively, for computing the generalised bisymmetric (containing symmetric solution as a special case) and skew-symmetric solutions of the generalised Sylvester matrix equation (including Sylvester and Lyapunov matrix equations as special cases) which is encountered in many systems and control applications. When the generalised Sylvester matrix equation has a unique generalised bisymmetric (skew-symmetric) solution, the first (second) iterative method converges to the generalised bisymmetric (skew-symmetric) solution of this matrix equation for any initial generalised bisymmetric (skew-symmetric) matrix. Finally, some numerical results are given to illustrate the effect of the theoretical results.

On the generalized bisymmetric and skew-symmetric solutions of the system of generalized Sylvester matrix equations

A matrix P is called a symmetric orthogonal matrix if P = P T = P −1. A matrix X is said to be a generalized bisymmetric with respect to P, if X = X T = PXP. It is obvious that every symmetric matrix is a generalized bisymmetric matrix with respect to I (identity matrix). In this article, we establish two iterative algorithms for solving the system of generalized Sylvester matrix equations (including the Sylvester and Lyapunov matrix equations as special cases) over the generalized bisymmetric and skew-symmetric matrices, respectively. When this system is consistent over the generalized bisymmetric (skew-symmetric) matrix Y, firstly it is demonstrated that the first (second) algorithm can obtain a generalized bisymmetric (skew-symmetric) solution for any initial generalized bisymmetric (skew-symmetric) matrix. Secondly, by the first (second) algorithm, we can obtain the least Frobenius norm generalized bisymmetric (skew-symmetric) solution for special initial generalized bisymmetric (skew-symmetric) matrices. Moreover, it is shown that the optimal approximate generalized bisymmetric (skew-symmetric) solution of this system for a given generalized bisymmetric (skew-symmetric) matrix can be derived by finding the least Frobenius norm generalized bisymmetric (skew-symmetric) solution of a new system of generalized Sylvester matrix equations. Finally, the iterative methods are tested with some numerical examples.

Inverse eigenvalue problems for bisymmetric matrices under a central principal submatrix constraint

This article considers an inverse eigenvalue problem for bisymmetric matrices under a central principal submatrix constraint and the corresponding optimal approximation problem. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we study a special form for the matrix of independent eigenvectors for a bisymmetric matrix. Based on these, we give some necessary and sufficient conditions for the solvability of the inverse eigenvalue problem, and we derive an expression for its general solution. Finally, we obtain an expression for the solution to the corresponding optimal approximation problem.

Least Squares Skew Bisymmetric Solution for a Kind of Quaternion Matrix Equation

In this paper, by using the Kronecker product of matrices and the complex representation of quaternion matrices, we discuss the special structure of quaternion skew bisymmetric matrices, and derive the expression of the least squares skew bisymmetric solution of the quaternion matrix equation AXB =C with the least norm.

The general coupled matrix equations over generalized bisymmetric matrices

In the present paper, by extending the idea of conjugate gradient (CG) method, we construct an iterative method to solve the general coupled matrix equations ∑ j = 1 p A ij X j B ij = M i , i = 1 , 2 , … , p , (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over generalized bisymmetric matrix group ( X 1 , X 2 , … , X p ) . By using the iterative method, the solvability of the general coupled matrix equations over generalized bisymmetric matrix group can be determined in the absence of roundoff errors. When the general coupled matrix equations are consistent over generalized bisymmetric matrices, a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm generalized bisymmetric solution group of the general coupled matrix equations can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation generalized bisymmetric solution group to a given matrix group ( X ^ 1 , X ^ 2 , … , X ^ p ) in Frobenius norm can be obtained by finding the least Frobenius norm generalized bisymmetric solution group of new general coupled matrix equations. The numerical results indicate that the iterative method works quite well in practice.