Left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal approximation problem

In this paper, the left and right inverse eigenpairs problem for generalized centrosymmetric matrices is considered. We obtain the necessary and sufficient conditions for the solvability of the problem, and we present the general expression of the solution. The related optimal approximation problem to a given matrix over the solution set is solved. In addition, a numerical algorithm and examples to solve the problem are given.     

Least‐squares solutions of matrix inverse problem for bi‐symmetric matrices with a submatrix constraint

An n × n real matrix A = (a_ij)_{n × n} is called bi‐symmetric matrix if A is both symmetric and per‐symmetric, that is, a_ij = a_ji and a_ij = a_n+1?1,n+1?i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least‐squares bi‐symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A^* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices

Based on the theory of inverse eigenvalue problem, a correction of an approximate model is discussed, which can be formulated as NX=XΛ, where X and Λ are given identified modal and eigenvalues matrices, respectively. The solvability conditions for a symmetric skew-Hamiltonian matrix N are established and an explicit expression of the solutions is derived. For any estimated matrix C of the analytical model, the best approximation matrix to minimize the Frobenius norm of C − N is provided and some numerical results are presented. A perturbation analysis of the solution is also performed, which has scarcely appeared in existing literatures. Supported by the National Natural Science Foundation of China(10571012, 10771022), the Beijing Natural Science Foundation (1062005) and the Beijing Educational Committee Foundation (KM200411232006, KM200611232010).     

Linear restriction problem of Hermitian reflexive matrices and its approximation

In this paper, we consider a linear restriction problem of Hermitian reflexive matrices and its approximation. By using the properties and structure of Hermitian reflexive matrices and the special properties of reflexive vectors and anti-reflexive vectors, we convert the linear restriction problem to an equivalence problem trickily, which is a special feature of this paper and is a different method from other articles. Then we solve this problem completely and also obtain its optimal approximate solution. Moreover, an algorithm provided for it and the numerical examples show that the algorithm is feasible.     

广义双对称矩阵反问题

利用矩阵的奇异值分解讨论了一类广义双对称矩阵反问题,得到了此类矩阵反问题有解的充要条件及通解的表达式.     

The inverse eigenvalue problem of generalized reflexive matrices and its approximation

This paper studies inverse eigenvalue problems of generalized reflexive matrices and their optimal approximations. Necessary and sufficient conditions for the solvability of the problems are derived, the solutions and their optimal approximations are provided.     

An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C

An iteration method is constructed to solve the linear matrix equation AXB=C over symmetric X. By this iteration method, the solvability of the equation AXB=C over symmetric X can be determined automatically, when the equation AXB=C is consistent over symmetric X, its solution can be obtained within finite iteration steps, and its least-norm symmetric solution can be obtained by choosing a special kind of initial iteration matrix, furthermore, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new matrix equation . Finally, numerical examples are given for finding the symmetric solution and the optimal approximation symmetric solution of the matrix equation AXB=C.     

线性流形上对称正交反对称矩阵反问题的最小二乘解

设P是n阶对称正交矩阵,如果n阶矩阵A满足AT=A和(PA)T=-PA,则称A为对称正交反对称矩阵,所有n阶对称正交反对称矩阵的全体记为SARnp.令S={A∈SARnp f(A)=‖AX-B‖=m in,X,B〗∈Rn×m本文讨论了下面两个问题问题Ⅰ给定C∈Rn×p,D∈Rp×p,求A∈S使得CTAC=D问题Ⅱ已知A~∈Rn×n,求A∧∈SE使得‖A~-A∧‖=m inA∈SE‖A~-A‖其中SE是问题Ⅰ的解集合.文中给出了问题Ⅰ有解的充要条件及其通解表达式.进而,指出了集合SE非空时,问题Ⅱ存在唯一解,并给出了解的表达式,从而得到了求解A∧的数值算法.     

The solvability conditions for inverse eigenproblem of symmetric and anti‐persymmetric matrices and its approximation

The problem of generating a matrix A with specified eigen‐pair, where A is a symmetric and anti‐persymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by ??????_Eⁿ. The optimal approximation problem associated with ??????_Eⁿ is discussed, that is: to find the nearest matrix to a given matrix A* by A∈??????_Eⁿ. The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix. Copyright © 2002 John Wiley & Sons, Ltd.     

广义次对称矩阵反问题的最小二乘解

讨论了广义次对称矩阵反问题的最小二乘解,得到了解的一般表达式,并就该问题的特殊情形:矩阵反问题,得到了可解的充分必要条件及解的通式.此外,证明了最佳逼近问题解的存在唯一性,并给出了其解的具体表达式.     

A modified quasi-boundary value method for solving the radially symmetric inverse heat conduction problem

In this paper, we consider a radially symmetric inverse heat conduction problem of determining the surface heat flux distribution from a fixed location inside a cylinder. This problem is ill-posed in the Hadamard sense and a conditional stability estimate is given for it. A modified quasi-boundary value regularization method is applied to formulate a regularized solution, and a sharp error estimate between the approximate solution and the exact solution is established by choosing a suitable regularization parameter. A numerical example is presented to verify the efficiency of the regularization method.     

Some decomposition results for companion matrices

Let C(p) denote the Frobenius companion matrix of the monic polynomial p with complex coefficients. The singular value decomposition and the QR-decomposition of C(p) are computed.     

Positive solutions for singular m-point boundary value problems with positive parameter

In this paper, some new existence results for positive solutions for a class of singular m-point boundary value problems with parameter be obtained.     

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.     

The left and right inverse eigenvalue problems of generalized reflexive and anti-reflexive matrices

Let n×n complex matrices R and S be nontrivial generalized reflection matrices, i.e., R^∗=R=R⁻¹≠±I_n, S^∗=S=S⁻¹≠±I_n. A complex matrix A with order n is said to be a generalized reflexive (or anti-reflexive ) matrix, if RAS=A (or RAS=−A). In this paper, the solvability conditions of the left and right inverse eigenvalue problems for generalized reflexive and anti-reflexive matrices are derived, and the general solutions are also given. In addition, the associated approximation solutions in the solution sets of the above problems are provided. The results in present paper extend some recent conclusions.     

Uncertain programming model for uncertain optimal assignment problem

This paper employs uncertain programming to deal with uncertain optimal assignment problem in which profit is uncertain. Within the framework of uncertain programming, it gives the uncertainty distribution of the optimal assignment profit, and the concept of α-optimal assignment for uncertain optimal assignment problem is proposed. Then α-optimal model is also constructed. Taking advantage of properties of uncertainty theory, α-optimal model can be transformed into a corresponding deterministic form, which can be solved by Kuhn–Munkres algorithm.     

Multiplicity results for second-order m-point boundary value problem

By using fixed-point theorems, some new results for multiplicity of positive solutions for a class of second-order m-point boundary value problem are obtained. The associated Green's function of the problem is also given.     

Existence and iteration of positive solutions for a generalized right-focal boundary value problem with p-Laplacian operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following third-order generalized right-focal boundary value problem with p-Laplacian operator:

     

The equivalence of curves in SL(n, R) and its application to ruled surfaces

The generating system of the differential algebra for invariant differential polynomials with two parametric curves is obtained. Conditions for the equivalence of two parametric curves families are given. We are also proved that the generating differential invariants of two parametric curves are independent. Finally, we reduce the SL(n, R)-equivalent problem for ruled surfaces to that of parametric curves.