Submatrix Constraint Research Articles

A Solution Matrix by IEVP under the Central Principle Submatrix Constraints

The n×n real matrix P is called centrosymmetric matrix if P=RPR, where R is permutation matrix with ones on cross diagonal (bottom left to top right) and zeroes elsewhere. In this article, the solvability conditions for left and right inverse eigenvalue problem (which is special case of inverse eigenvalue problem) under the submatrix constraint for generalized centrosymmetric matrices are derived, and the general solution is also given. In addition, we provide a feasible algorithm for computing the general solution, which is proved by a numerical example.

Matrix Inverse Problemunder the Central Master Submatrix Constraint AX = B of the Generalized Centersymmetric Solution

This paper begins by identifying a problem related to the solvability of the matrix equation AX=B, particularly within the realm of generalized centrosymmetric matrices. It exploits the distinctive properties inherent in generalized centrosymmetric matrices, primary focus involves investigating their solutions within the context of the matrix equation AX=B. This paper establishes the necessary and sufficient conditions for solvability while formulating a comprehensive expression for the generalized centrosymmetric solution. Furthermore, the paper delves into addressing the associated optimal approximation quandary concerning a specified matrix within the solution set.

Conjugate gradient-like algorithms for constrained operator equation related to quadratic inverse eigenvalue problems

As is well known, linear operator equations have wide applications in many areas of engineering and applied mathematics. In the present paper, we are interested in solving the linear operator equation $$\begin{aligned} {\mathscr {F}}(J)+{\mathscr {G}}(K)+{\mathscr {H}}(L)=N, \end{aligned}$$ where J, K and L should be partially bisymmetric under a prescribed submatrix constraint. Three conjugate gradient-like algorithms are derived for solving this constrained operator equation including the Lyapunov, Stein and Sylvester matrix equations and the quadratic inverse eigenvalue problem as special cases. The algorithms converge to the solutions of the linear operator equation within a finite number of iterations in the absence of round-off errors. At the end, the accuracy and efficiency of the introduced algorithms are demonstrated numerically with three examples.

Matrix LSQR algorithms for solving constrained quadratic inverse eigenvalue problem

The inverse eigenvalue problem appears in many applications such as control design, seismic tomography, exploration and remote sensing, molecular spectroscopy, particle physics, structural analysis, and mechanical system simulation. This paper investigates the matrix form of LSQR methods for solving the quadratic inverse eigenvalue problem with partially bisymmetric matrices under a prescribed submatrix constraint. In order to illustrate the effectiveness and feasibility of our results, one numerical example is presented.

Submatrix constrained left and right inverse eigenvalue problem for centrosymmetric matrices

ABSTRACT In this article, we will find centrosymmetric matrix solutions A of the left and right inverse eigenvalue problem under a submatrix constraint, where is also a centrosymmetric matrix. In other words, expand the system (matrix) A from the centre subsystem (submatrix) satisfying the matrix constraint, where A and are both centrosymmetric matrices. Using the similar structure of A and , we discuss the sufficient and necessary conditions for the left and right inverse eigenvalue problem having solutions, and give the expression for its general solution. Then, we discuss its optimal approximation problem and gain the expression of its solution. Last, we provide a feasible algorithm for computing the unique solution to its optimal approximation problem, which is proved by some numerical examples.

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.

Iterative method to the coupled operator matrix equations with sub-matrix constraint and its application in control

A finite iterative algorithm is presented for solving the numerical solutions to the coupled operator matrix equations in Zhang (2017b). In this paper, a new finite iterative algorithm is presented for solving the constraint solutions to the coupled operator matrix equations [Formula: see text], where the constraint solutions include symmetric solutions, bisymmetric solutions and reflexive solutions as special cases. If this system is consistent, for any initial constraint matrices, the exact constraint solutions can be obtained by the introduced algorithm within finite iterative steps in the absence of the roundoff errors. Also, if this system is not consistent, the least-norm constraint solutions can be obtained within the finite iteration steps in the absence of the roundoff errors. Furthermore, if a group of suitable matrices are given, the optimal approximation solutions can be derived. Finally, several numerical examples are given to show the effectiveness of the presented iterative algorithm.

Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix

Abstract Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

The generalized conjugate direction method for solving quadratic inverse eigenvalue problems over generalized skew Hamiltonian matrices with a submatrix constraint

In this paper, we consider a class of constrained quadratic inverse eigenvalue Problem 1.1. Then, a generalized conjugate direction method is proposed to obtain the generalized skew Hamiltonian matrix solutions with a submatrix constraint. In addition, by choosing a special kind of initial matrices, it is shown that the unique least Frobenius norm solutions can be obtained consequently. Some numerical results are reported to demonstrate the efficiency of our algorithm.

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.

Solving constrained quadratic inverse eigenvalue problem via conjugate direction method

Inverse eigenvalue problems are among the most important problems in numerical linear algebra. This paper is concerned with the problem of designing an iterative method for a quadratic inverse eigenvalue problem of the form MXΛ2+GXΛ+KX=0 where M, G and K should be partially doubly symmetric under a prescribed submatrix constraint. This kind of quadratic inverse eigenvalue problem includes the classical and generalized inverse eigenvalue problems as special cases. We propose the conjugate direction (CD) method for computing the least Frobenius norm solutions of this constrained quadratic inverse eigenvalue problem in a finite number of steps in the absence of round-off errors. The CD method can automatically determine the solvability of this problem. Finally, the overall performance and efficiency of the CD method for computing approximate solutions of the problem are discussed.

An efficient algorithm based on Lanczos type of BCR to solve constrained quadratic inverse eigenvalue problems

This paper proposes the Lanczos type of biconjugate residual (BCR) algorithm for solving the quadratic inverse eigenvalue problem L 2 X Λ 2 + L 1 X Λ + L 0 X = 0 where L 2 , L 1 and L 0 should be partially bisymmetric under a prescribed submatrix constraint. An analysis reveals that the algorithm obtains the solutions of the constrained quadratic inverse eigenvalue problem in finitely many steps in the absence of round-off errors. Finally numerical results are performed to confirm the analysis and to illustrate the efficiency of the proposed algorithm.

An iterative method for obtaining the Least squares solutions of quadratic inverse eigenvalue problems over generalized Hamiltonian matrix with submatrix constraints

In this paper, we consider a class of constrained matrix quadratic inverse eigenvalue problem and its optimal approximation problem. It is proved that the proposed algorithm always converge to the generalized Hamiltonian solutions with a submatrix constraint of Problem 1.1 within finite iterative steps in the absence of roundoff error. In addition, by choosing a special kind of initial matrices, it is shown that the minimum norm solution of Problem 1.1 can be obtained consequently. At last, for a given matrix group in the solution set of Problem 1.1, it is proved that the unique optimal approximation solution of Problem 1.2 can be also obtained. Some numerical results are reported to demonstrate the efficiency of our algorithm.

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.

Least-squares solutions of generalized inverse eigenvalue problem over Hermitian–Hamiltonian matrices with a submatrix constraint

In this paper, a gradient-based iterative algorithm is proposed for finding the least-squares solutions of the following constrained generalized inverse eigenvalue problem: given \(X\in C^{n\times m}\), \(\Lambda =\mathrm{diag}(\lambda _1,\lambda _2,\ldots ,\lambda _m)\in C^{m\times m}\), find \(A^*,B^*\in C^{n\times n}\), such that \(\Vert AX-BX\Lambda \Vert \) is minimized, where \(A^*,B^*\) are Hermitian–Hamiltonian except for a special submatrix. For any initial constrained matrices, a solution pair \((A^*,B^*)\) 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. A numerical example is given to show the efficiency of the proposed algorithm.

Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction

AbstractGeneralised centrohermitian and skew-centrohermitian matrices arise in a variety of applications in different fields. Based on the vibrating structure equation where M, D, G, K are given matrices with appropriate sizes and x is a column vector, we design a new vibrating structure mode. This mode can be discretised as the left and right inverse eigenvalue problem of a certain structured matrix. When the structured matrix is generalised centrohermitian, we discuss its left and right inverse eigenvalue problem with a submatrix constraint, and then get necessary and sufficient conditions such that the problem is solvable. A general representation of the solutions is presented, and an analytical expression for the solution of the optimal approximation problem in the Frobenius norm is obtained. Finally, the corresponding algorithm to compute the unique optimal approximate solution is presented, and we provide an illustrative numerical example.

Comments on “The reflexive least squares solutions of the general coupled matrix equations with a submatrix constraint”

In this note, a technical error is pointed out in the equality (2) of Lemma 2.2 in the above mentioned paper, and the corresponding correction is presented.

The inverse eigenproblem with a submatrix constraint and the associated approximation problem for [formula omitted]-symmetric matrices

Let R∈Rn×n and S∈Rn×n be nontrivial involutions, i.e., R=R−1≠±I and S=S−1≠±I. A matrix A∈Rn×n is called (R,S)-symmetric if RAS=A. This paper presents a (R,S)-symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint. The solvability condition of the constrained inverse eigenproblem is also derived. The existence, the uniqueness and the expression of the (R,S)-symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved, respectively. An algorithm is presented to compute the (R,S)-symmetric matrix solution to the best approximation problem. Two numerical examples are given to illustrate the effectiveness of our results.

The reflexive least squares solutions of the general coupled matrix equations with a submatrix constraint

In this paper, we construct an iterative method to solve the general coupled matrix equations∑j=1lAijXjBij=Ci,i=1,2,…,t,where Xj∈Rnj×nj(j=1,2,…,l) is a reflexive matrix with a specified central principal submatrix. The algorithm produces suitable [X1,X2,…,Xl] such that ∑i=1t∑j=1lAijXjBij-Ci=min within finite iteration steps in the absence of roundoff errors. We show that the algorithm is stable any case. The algorithm requires little storage capacity. Given numerical examples show that the algorithm is efficient.

An efficient algorithm for the submatrix constraint of the matrix equation A 1 X 1 B 1+A 2 X 2 B 2+···+A l X l B l =C

Matrix A=(a ij )∈R n×n is said to be bisymmetric if a ij =a ji =a n+1−j, n+1−i for all 1≤i, j≤n. In this paper, an efficient algorithm is presented for minimizing ‖A 1 X 1 B 1+A 2 X 2 B 2+···+A l X l B l −C‖, where ‖·‖ is the Frobenius norm and is bisymmetric with a specified central principal submatrix . The algorithm produces suitable [X 1, X 2, …, X l ] such that ‖A 1 X 1 B 1+A 2 X 2 B 2+···+A l X l B l −C‖=min within finite iteration steps in the absence of roundoff errors. The results of given numerical experiments show that the algorithm has fast convergence rate.