Musheng Wei

A new structure-preserving method for quaternion Hermitian eigenvalue problems

In this paper we propose a novel structure-preserving algorithm for solving the right eigenvalue problem of quaternion Hermitian matrices. The algorithm is based on the structure-preserving tridiagonalization of the real counterpart for quaternion Hermitian matrices by applying orthogonal JRS-symplectic matrices. The algorithm is numerically stable because we use orthogonal transformations; the algorithm is very efficient, it costs about a quarter arithmetical operations, and a quarter to one-eighth CPU times, comparing with standard general-purpose algorithms. Numerical experiments are provided to demonstrate the efficiency of the structure-preserving algorithm.

Real structure-preserving algorithms of Householder based transformations for quaternion matrices

In this paper, we survey three different forms of Householder based transformations for quaternion matrices in the literature, and propose a new form of quaternion Householder based transformation. We propose real structure-preserving algorithms of these Householder based transformations, which make the procedure computationally more flexible and efficient. We compare the computation counts and assignment numbers of these algorithms. We also compare the effectiveness of these real structure-preserving algorithms applying to the quaternion QRD and the quaternion SVD.All these four real structure-preserving algorithms are more efficient, comparing to the algorithms which apply Quaternion Toolbox using quaternion arithmetics, or algorithms which directly performs real Householder transformations on the real representation of a quaternion matrix. Among these four real structure-preserving algorithms, the most efficient ones are based on quaternion Householder reflection, and new proposed Householder based transformation.

Perturbation analysis for the matrix least squares problem A X B = C

Let S and S ? be two sets of solutions to matrix least squares problem (LSP) A X B = C and the perturbed matrix LSP A ? X ? B ? = C ? , respectively, where A ? = A + Δ A , B ? = B + Δ B , C ? = C + Δ C , and Δ A , Δ B , Δ C are all small perturbation matrices. For any given X ? S , we deduce general formulas of the least squares solutions X ? ? S ? that are closest to X under appropriated norms, meanwhile, we present the corresponding distances between them. With the obtained results, we derive perturbation bounds for the nearest least squares solutions. At last, a numerical example is provided to verify our analysis.

SYMMETRIC INVERSE GENERALIZED EIGENVALUE PROBLEM WITH SUBMATRIX CONSTRAINTS IN STRUCTURAL DYNAMIC MODEL UPDATING

In this literature, the symmetric inverse generalized eigenvalue problem with subma- trix constraints and its corresponding optimal approximation problem are studied. A necessary and sufficient condition for solvability is derived, and when solvable, the general solutions are presented.

The minimal norm least squares Hermitian solution of the complex matrix equation AXB+CXD=E

Abstract In this paper, by applying the real representations of complex matrices, the particular structure of the real representations and the Moore–Penrose generalized inverse, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation A X B + C X D = E . And we also derive the minimal norm least squares Hermitian solution of the complex matrix equation A X B = E . Our proposed formulas only involve real matrices, and therefore are more effective and portable than those reported in Yuan and Liao (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of complex matrices. Two numerical examples are provided to demonstrate the effectiveness of our algorithms.

A new real structure-preserving quaternion QR algorithm

Abstract New real structure-preserving decompositions are introduced to develop fast and robust algorithms for the (right) eigenproblem of general quaternion matrices. Under the orthogonally J R S -symplectic transformations, the Francis J R S -QR step and the J R S -QR algorithm are firstly proposed for J R S -symmetric matrices and then applied to calculate the Schur form of quaternion matrices. A novel quaternion Givens matrix is defined and utilized to compute the QR factorization of quaternion Hessenberg matrices. An implicit double shift quaternion QR algorithm is presented with a technique for automatically choosing shifts and within real operations. Numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms.

Minimum rank (skew) Hermitian solutions to the matrix approximation problem in the spectral norm

In this paper, we discuss the following two minimum rank matrix approximation problems in the spectral norm: (i)For given A = A H ? C m i? m , B ? C m i? n , determining X ? S 1 , such that rank ( X ) = min Y ? S 1 rank ( Y ) , S 1 = { Y = Y H ? C n i? n : ? A - B Y B H ? 2 = min } .(ii)For given A = - A H ? C m i? m , B ? C m i? n , determining X ? S 2 , such that rank ( X ) = min Y ? S 2 rank ( Y ) , S 2 = { Y = - Y H ? C n i? n : ? A - B Y B H ? 2 = min } .By applying the norm-preserving dilation theorem, the Hermitian-type (skew-Hermitian-type) generalized singular value decomposition (HGSVD, SHGSVD), we characterize the expressions of the minimum rank and derive a general form of minimum rank (skew) Hermitian solutions to the matrix approximation problem.

On the power method for quaternion right eigenvalue problem

Abstract In this paper, we study the power method of the right eigenvalue problem of a quaternion matrix A . If A is Hermitian, we propose the power method that is a direct generalization of that of complex Hermitian matrix. When A is non-Hermitian, by applying the properties of quaternion right eigenvalues, we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector. We also briefly discuss the inverse power method and shift inverse power method for the both cases. The real structure-preserving algorithm of the power method in the two cases are also proposed, and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method.

An efficient method for special least squares solution of the complex matrix equation (AXB,CXD)=(E,F)

Abstract In this paper, we propose an efficient method for special least squares solution of the complex matrix equation ( A X B , C X D ) = ( E , F ) . By using the real representation matrices of complex matrices, the particular structure of the real representation matrices, the Moore–Penrose generalized inverse and the Kronecker product, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation ( A X B , C X D ) = ( E , F ) , which was studied by a product of matrices and vectors in Wang et al. (2016). Our resulting formulas only involve real matrices, and the corresponding algorithm only performs real arithmetic. Therefore our proposed method is more effective and portable. Finally, we give three numerical examples to illustrate the effectiveness of our proposed method.