Zhibing Liu

The inverse eigenvalue problem of generalized reflexive matrices and its approximation

A real symmetric unipotent matrix P is said to be generalized reflection matrix. A real matrix A is said to be a generalized reflexive matrix with respect to generalized reflection matrix dual (P,Q) if A = PAQ. This paper involves related inverse eigenvalue problems of generalized reflexive matrices and their optimal approximation. Necessary and sufficient conditions for the solvability of the problem are derived,the general expression of the solution is given. The optimal approximate solution is also provided.

An Inverse Eigenvalue Problem for Symmetric Arrow-Plus-Jacobi Matrices

In this paper we study a kind of inverse eigenvalue problem for a special kind of real symmetric matrices: the real symmetric Arrow-plus-Jacobi matrices. That is, matrices which look like arrow matrices forward and Jacobi backward, from the( p, p )station, 1 ≤p ≤ n. We give a necessary and sufficient condition for the existence of such a matrix. Our results are constructive, in the sense that they generate an algorithmic procedure to construct the matrix.

Existence and Construction of Nonnegative Matrices with Pure Image Spectrum

The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers σ to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list except for one (the Perron eigenvalue) are pure image numbers. Lets. Let σ = (ρ,b₁i, b₁i,···,b_ki, b_ki)be a list of complex numbers with ρ,bj >; 0 for j =1,2,···,k . A simple necessary and sufficient conditions for the existence of an entry wise nonnegative 2k +1 order matrix A with spectrum σ are presented, and the proof is elementary.

The Inverse Eigenvalue Problem of Generalized Reflexive Matrices

A real symmetric unipotent matrix P is said to be n×n generalized reflection matrix. A real matrix A is said to be a generalized reflexive matrix with respect to generalized reflection matrix dual (P, Q) if A = PAQ. This paper involves related inverse eigenvalue problems of generalized reflexive matrices and their optimal approximation. Necessary and sufficient conditions for the solvability of the problem are derived, the general expression of the solution is given. The optimal approximate solution is also provided.

Singular value decomposition for central extended matrix

We study the Singular value decomposition of a special class of centrosymmetric matrices: the central extended matrix which is not only a centrosymmetric matrix but also a row extended matrix. The formula for the Singular value decomposition factorization is obtained, and the relationship of the central extended matrix with mother matrix is also obtained.

The generalized reflexive solutions of the matrix equation AX = B

In this paper, we consider the generalized reflexive solutions of the matrix equation AX = B. The necessary and sufficient conditions for the solvability of the problem are given, moreover, a simple and eigenvector-free formula of the general solution to the matrix equation is presented using Moore-Penrose generalized inverses of the coefficient matrix A. In addition, in corresponding solution set of the equation, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been provided.

The inverse eigenvalue problem of generalized anti-reflexive matrices

A real symmetric unipotent matrix P is said to be generalized reflection matrix. A real matrix A is said to be a generalized anti-reflexive matrix with respect to generalized reflection matrix dual (P,Q) if A =-PAQ. This paper involves related inverse eigenvalue problems of generalized antire flexive matrices and their optimal approximation. Necessary and sufficient conditions for the solvability of the problem are derived, the general expression of the solution is given. The optimal approximate solution is also provided.

On the Construction of Positive Definite Doubly Arrow Matrix from Two Eigenpairs

A class of inverse eigenvalue problem is proposed for real symmetric positive definite Arrow matrices. Necessary and sufficient conditions for the existence of a unique solution of this problem, as well as the analytic formula of this solution are derived, Our results are on structive, in the sense that they generate an algorithmic procedure to construct the matrix.

The Wiener-Askey Polynomial Chaos for Diffusion Problems with Uncertainty

Efficient methods based on Galerkin projections and the Wiener-Askey polynomial chaos are constructed to solve the steady/unsteady state diffusion problems with uncertainty. Some numerical examples are used to confirm the efficiency of the methods. The exponential convergence of the methods is demonstrated for each problem.

On the Construction of Positive Definite Arrow Matrix from Two Eigenpairs

A class of inverse eigenvalue problem is proposed for real symmetric positive definite Arrow matrices. Necessary and sufficient conditions for the existence of a unique solution of this problem, as well as the analytic formula of this solution are derived.