On solutions of matrix equations and

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BW, with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.     

Sign patterns for eigenmatrices of nonnegative matrices

For a square (0,?1,??1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this article, we investigate the relationship between sign patterns and matrices that diagonalize an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalizes some irreducible nonnegative matrix. Further, we characterize the sign patterns S such that some member of Q(S) diagonalizes an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realized as the spectrum of an irreducible nonnegative matrix M that is diagonalized by a matrix in the qualitative class of some S ² NS sign pattern.     

一类四元数矩阵方程的循环解及其最佳逼近

本文研究了四元数体上矩阵方程XB = C 的循环解及其最佳逼近问题. 利用循环矩阵的结构表示式, 以及四元数矩阵的复分解, 得到了方程XB = C 的循环解存在条件及其通解形式; 在循环矩阵约束条件下, 给出了该方程的最小二乘解集合; 与此同时, 在最小二乘解集合中, 获得与给定四元数循环矩阵的最佳逼近解. 推广了约束矩阵方程的数值求解范围. 数值算例验证了本文算法的可行性.     

一类四元数矩阵方程的循环解及其最佳逼近

本文研究了四元数体上矩阵方程XB=C的循环解及其最佳逼近问题.利用循环矩阵的结构表示式,以及四元数矩阵的复分解,得到了方程XB=C的循环解存在条件及其通解形式;在循环矩阵约束条件下,给出了该方程的最小二乘解集合;与此同时,在最小二乘解集合中,获得与给定四元数循环矩阵的最佳逼近解.推广了约束矩阵方程的数值求解范围.数值算例验证了本文算法的可行性.     

BIORTHOGONAL UNCONDITIONAL BASES OF COMPACTLY SUPPORTED MATRIX VALUED WAVELETS

In this paper, we introduce the concept of biorthogonal matrix valued wavelets. We elaborate on perfect reconstruction matrix filter banks which are assembled by matrix FIR fllters and we deduce that the resulting matrix valued wavelet functions have compact support. Moreover, we form biorthogonal unconditional bases for the space of matrix valued signals. To validate the theory, a class of biorthogonal and orthonormal matrix valued wavelets are given. The connection of the present scheme with the theory of multiwavelets are also explored.     

The <Emphasis Type="Italic">k</Emphasis>-Fibonacci matrix and the Pascal matrix

We define the k-Fibonacci matrix as an extension of the classical Fibonacci matrix and relationed with the k-Fibonacci numbers. Then we give two factorizations of the Pascal matrix involving the k-Fibonacci matrix and two new matrices, L and R. As a consequence we find some combinatorial formulas involving the k-Fibonacci numbers.     

The perturbed laplacian matrix of a graph

For a graph G, we define its perturbed Laplacian matrix as D?A(G) where A(G) is the adjacency matrix of G and D is an arbitrary diagonal matrix. Both the Laplacian matrix and the negative of the adjacency matrix are special instances of the perturbed Laplacian. Several well-known results, contained in the classical work of Fiedler and in more recent contributions of other authors are shown to be true, with suitable modifications, for the perturbed Laplacian. An appropriate generalization of the monotonicity property of a Fiedler vector for a tree is obtained. Some of the results are applied to interval graphs.     

On -Hadamard matrices

In this paper, we use the Sylvester's approach to construct another Hadamard matrix, namely a J_m-Hadamard matrix, from a given one. Consequently, we can generate other 2^m-1 Hadamard matrices from the constructed J_m-Hadamard matrix. Finally, we also discuss the Kronecker product of an Hadamard matrix and a J_m-Hadamard matrix.     

Function of a square matrix with repeated eigenvalues

An analytical function f(A) of an arbitrary n×n constant matrix A is determined and expressed by the “fundamental formula”, the linear combination of constituent matrices. The constituent matrices Z_kh, which depend on A but not on the function f(s), are computed from the given matrix A, that may have repeated eigenvalues. The associated companion matrix C and Jordan matrix J are then expressed when all the eigenvalues with multiplicities are known. Several other related matrices, such as Vandermonde matrix V, modal matrix W, Krylov matrix K and their inverses, are also derived and depicted as in a 2-D or 3-D mapping diagram. The constituent matrices Z_kh of A are thus obtained by these matrices through similarity matrix transformations. Alternatively, efficient and direct approaches for Z_kh can be found by the linear combination of matrices, that may be further simplified by writing them in “super column matrix” forms. Finally, a typical example is provided to show the merit of several approaches for the constituent matrices of a given matrix A.     

ON NEAR-RINGS WITH MATRIX UNITS

Abstract

In ring theory it is well known that a ring R with identity is isomorphic to a matrix ring if and only if R has a set of matrix units. In this paper, the above result is extended to matrix near-rings and it is proved that a near-ring R with identity is isomorphic to a matrix near-ring if and only if R has a set of matrix units and satisfies two other conditions. As a consequence of this result several examples of matrix near-rings are given and for a finite group (Γ, +) with o(Γ) > 2 it is proved that M₀ (Γⁿ) is (isomorphic to) a matrix near-ring.     

How to use RSVD to solve the matrix equation A=BXC

Through the restricted singular value decomposition (RSVD) of the matrix triplet (C, A, B), we show in this note how to choose a variable matrix X such that the matrix pencil A ? BXC attains its maximal and minimal ranks. As applications, we show how to use the RSVD to solve the matrix equation A = BXC.     

Hermitian‐type generalized singular value decomposition with applications

For a pair of given Hermitian matrix A and rectangular matrix B with the same row number, we reformulate a well‐known simultaneous Hermitian‐type generalized singular value decomposition (HGSVD) with more precise structure and parameters and use it to derive some algebraic properties of the linear Hermitian matrix function A?BXB* and Hermitian solution of the matrix equation BXB* = A, and the canonical form of a partitioned Hermitian matrix and some optimization problems on its inertia and rank. Copyright © 2012 John Wiley & Sons, Ltd.     

Matrix-free preconditioning using partial matrix estimation

We consider matrix-free solver environments where information about the underlying matrix is available only through matrix vector computations which do not have access to a fully assembled matrix. We introduce the notion of partial matrix estimation for constructing good algebraic preconditioners used in Krylov iterative methods in such matrix-free environments, and formulate three new graph coloring problems for partial matrix estimation. Numerical experiments utilizing one of these formulations demonstrate the viability of this approach. AMS subject classification (2000) 65F10, 65F50, 49M37, 90C06     

Symmetric-Triangular Decomposition and its Applications Part I: Theorems and Algorithms

A new decomposition of a nonsingular matrix, the Symmetric times Triangular (ST) decomposition, is proposed. By this decomposition, every nonsingular matrix can be represented as a product of a symmetric matrix S and a triangular matrix T. Furthermore, S can be made positive definite. Two numerical algorithms for computing the ST decomposition with positive definite S are presented.     

An efficient iterative method for solving the matrix equation AXB + CYD = E

This paper presents an iterative method for solving the matrix equation AXB + CYD = E with real matrices X and Y. By this iterative method, the solvability of the matrix equation can be determined automatically. And when the matrix equation is consistent, then, for any initial matrix pair [X₀, Y₀], a solution pair can be obtained within finite iteration steps in the absence of round‐off errors, and the least norm solution pair can be obtained by choosing a special kind of initial matrix pair. Furthermore, the optimal approximation solution pair to a given matrix pair [X?, ?] in a Frobenius norm can be obtained by finding the least norm solution pair of a new matrix equation AX?B + C?D = ?, where ? = E ? AX?B ? C?D. The given numerical examples show that the iterative method is efficient. Copyright © 2005 John Wiley & Sons, Ltd.     

Properties of Hadamard product of inverse M‐matrices

This paper concerns with the properties of Hadamard product of inverse M‐matrices. Structures of tridiagonal inverse M‐matrices and Hessenberg inverse M‐matrices are analysed. It is proved that the product A ○ A^T satisfies Willoughby's necessary conditions for being an inverse M‐matrix when A is an irreducible inverse M‐matrix. It is also proved that when A is either a Hessenberg inverse M‐matrix or a tridiagonal inverse M‐matrix then A ○ A^T is an inverse M‐matrix. Based on these results, the conjecture that A ○ A^T is an inverse M‐matrix when A is an inverse M‐matrix is made. Unfortunately, the conjecture is not true. Copyright © 2004 John Wiley Sons, Ltd.     

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.     

A method for computing the Perron root for primitive matrices

Following the Perron theorem, the spectral radius of a primitive matrix is a simple eigenvalue. It is shown that for a primitive matrix A, there is a positive rank one matrix X such that B = A ° X , where ° denotes the Hadamard product of matrices, and such that the row (column) sums of matrix B are the same and equal to the Perron root. An iterative algorithm is presented to obtain matrix B without an explicit knowledge of X. The convergence rate of this algorithm is similar to that of the power method but it uses less computational load. A byproduct of the proposed algorithm is a new method for calculating the first eigenvector.     

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.     

Gyroscopic Stabilization of 2nd-Order-Systems with Indefinite Damping

We consider the gyroscopic stabilization of the unstable system ẍ + D ẋ + Kx = 0 with positive definite stiffness matrix K. The indefinite damping matrix D is responsible for the instability of the system. The modelling of sliding bearings can lead to negative damping, see [6]. A gyroscopic stabilization of an unstable mechanical system with indefinite damping matrix was investigated in [4] in the case of matrix order n = 2 using the Routh-Hurwitz criterion. The question was raised whether an unstable system can be stabilized by adding a gyroscopic term Gẋ with a suitable skew-symmetric matrix G = −G^T . (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)