The Monotonicity Problems for Generalized Inverses of Matrices in H (n, ≥)

On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.

THE DRAZIN INVERSE OF MATRICES OF QUATERNIONS

Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let Hm×n denote the set of all m×n matrices over H.If A=(ars)∈Hm×n,then there exist A1 and A2∈Cm×n such that A=A1+A2j.Let AC denote the complexrepresentation of A,that is the 2m×2n complex matrix Ac=（（A1/A2）（-A2/A1））（see[1,2]）.We denote by AD the Drazin inverse of A∈Hm×n which is the unique solution of the e-

The Explicit Expressions of Level- k(r1 ,r2 ,… ,rk) Circulant Matrices and Their Some Properties

In this paper,we give the explicit expressions of level k (r 1,r 2,…,r k) circulant matrices of order n 1n 2…n k,and the explicit expressions for the eigenvalues,the determinants and the inverse matrices of the kind level k (r 1,r 2,…,r k) circulant matrices are derived,and it is also proved that the sort of matrices are diagonalizable.

The Explicit Expressions of Level-k Circulant Matrices of Type(n_1,n_2.…,n_k) and Their Some Properties

In this paper, we give the explicit expressions of level-k circulant matrices of type (n1,n2,…nk) and of order n1n2…nk,and the explicit expressions for the eigenvalues,the determinants and the inverse matrices of the kind level-k circulant matrices are derived,and it is also proved that the sort matrices are unitarily diagonalizable.

REVERSE ORDER LAW FOR GROUP INVERSE OF PRODUCT OF MATRICES ON SKEW FIELDS

Let Kn×n be the set of all n×n matrices and Krn×n the set {A∈Kn×n|rankA=r} on askew field K. Zhuang [1] denotes by A# the group inverse of A∈Kn×n which is the solu-tion of the euqations:AXA=A, XAX=X, AX=AX.

New Approach for the Inversion of Structured Matrices via Newton’s Iteration

Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ?rapidly refines a crude initial approximation X0?to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this method to n× n?structured matrices M?, in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster computations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.

Solubility Existence of Inverse Eigenvalue Problem for a Class of Singular Hermitian Matrices

In this article,we discuss singular Hermitian matrices of rank greater or equal to four for an inverse eigenvalue problem.Specifically,we look into how to generate n by n singular Hermitian matrices of ranks four and five from a prescribed spectrum.Numerical examples are presented in each case to illustrate these scenarios.It was established that given a prescribed spectral datum and it multiplies,then the solubility of the inverse eigenvalue problem for n by n singular Hermitian matrices of rank r exists.

Construction of protograph LDPC codes with circular generator matrices

The application of protograph low density parity check （LDPC） codes involves the encoding complexity problem. Since the generator matrices are dense, and if the positions of ＂1＂ s are irregularity, the encoder needs to store every ＂1＂ of the generator matrices by using huge chip area. In order to solve this problem, we need to design the protograph LDPC codes with circular generator matrices. A theorem concerning the circulating property of generator matrices of nonsingular protograph LDPC codes is proposed. The circulating property of generator matrix of nonsingular protograph LDPC codes can be obtained from the corresponding quasi-cyclic parity check matrix. This paper gives a scheme of constructing protograph LDPC codes with circulating generator matrices, and it reveals that the fast encoding algorithm of protograph LDPC codes has lower encoding complexity under the condition of the proposed theorem. Simulation results in ad- ditive white Gaussian noise （AWGN） channels show that the bit error rate （BER） performance of the designed codes based on the proposed theorem is much better than that of GB20600 LDPC codes and Tanner LDPC codes.

Inverse Eigenvalue Problem for Generalized Arrow-Like Matrices

This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided.

NEW CONSTRUCTIONS OF LDPC CODES BASED ON CIRCULANT PERMUTATION MATRICES FOR AWGN CHANNEL

Two new design approaches for constructing Low-Density Parity-Check(LDPC) codes are proposed.One is used to design regular Quasi-Cyclic LDPC(QC-LDPC) codes with girth at least 8.The other is used to design irregular LDPC codes.Both of their parity-check matrices are composed of Circulant Permutation Matrices(CPMs).When iteratively decoded with the Sum-Product Algorithm(SPA),these proposed codes exhibit good performances over the AWGN channel.

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

is gained by deleting the kth row and the kth column （k=1,2,...,n） from Tn.We put for-ward an inverse eigenvalue problem to be that:If we don’t know the matrix T1,n,but weknow all eigenvalues of matrix T1,k-1,all eigenvalues of matrix Tk+1,k,and all eigenvaluesof matrix T1,n could we construct the matrix T1,n.Let μ1,μ2,…,μk-1,μk,μk+1,…,μn-1,

BEST APPROXIMATION BY NORMAL MATRICES WITH SPECTRAL CONSTRAINTS

The problem of best approximating, a given square complex matrix in the Frobenius norm by normal matrices under a given spectral restriction is considered. The ne cessary and sufficient condition for the solvability of the problem is given. A numerical algorithm for solving the problem is provided and a numerical example is presented.

Design of good QC-LDPC codes without small girth in the p-plane

A construction method based on the p-plane to design high-girth quasi-cyclic low-density parity-check （QC-LDPC） codes is proposed. Firstly the good points in every line of the p-plane can be ascertained through filtering the bad points, because the designed parity-check matrixes using these points have the short cycles in Tanner graph of codes. Then one of the best points from the residual good points of every line in the p-plane will be found, respectively. The optimal point is also singled out according to the bit error rate （BER） performance of the QC-LDPC codes at last. Explicit necessary and sufficient conditions for the QC-LDPC codes to have no short cycles are presented which are in favor of removing the bad points in the p-plane. Since preventing the short cycles also prevents the small stopping sets, the proposed construction method also leads to QC-LDPC codes with a higher stopping distance.

The Rank of Rank-2 Modified Matrix

In this paper, the authors discuss the relationship in detail between the rank of M in the modified matrix M = A ＋ BC^＊ and the rank of matrix A. The authors do believe the results are useful tools in the modified matrices.

Type Synthesis of Fully-Decoupled 3R2T Parallel Mechanisms Based on Driven-Chain Principle

To avoid the existence of nonlinear and strong coupling in parallel mechanisms,it is necessary to address special care to the type synthesis of mechanisms,especially for the type synthesis of fully-decoupled parallel mechanisms. Based on the screw theory and the driven-chain principle,a methodology of structural synthesis for fully-decoupled three-rotational( 3R) and two-translational( 2T)parallel mechanisms was proposed by analyzing the characteristics of the input-output relations for fully-decoupled parallel mechanisms.Firstly,according to the desired kinematic characteristics of fullydecoupled parallel mechanisms,a method was presented by virtue of screw theory to synthesize the desired forms for both the direct and the inverse Jacobian matrices. Secondly,according to the feature of the direct and the inverse Jacobian matrices,the effective screws,the actuated screws and the mobile un-actuated screws of each leg were established based on the reciprocal screw theory and all possible topology structures fulfilling the requirements were obtained.Finally,the desired fully-decoupled parallel mechanisms could be synthesized by using the structural synthesis rule and structural synthesis of fully-decoupled 3R2 T parallel mechanisms could be obtained exploiting the abovementioned methodology. Furthermore,the Jacobian matrix of a synthesized 3R2 T parallel mechanism was deduced to demonstrate the decoupling feature of the parallel mechanism,which also validated the correctness of the methodology of the type synthesis for fully-decoupled 3R2 T parallel mechanisms.

THE SOLVABILITY CONDITIONS FOR THE INVERSE PROBLEM OF MATRICES POSITIVE SEMIDEFINITE ON A SUBSPACE

This paper studies the following two problems: Problem I. Given X, B is-an-element-of R(n x m), find A is-an-element-of P(s,n), such that AX = B, where Ps, n = {A is-an-element-of SR(n x n)\x(T) Ax greater-than-or-equal-to 0, for-all S(T) x = 0, for given S is-an-element-of R(p)n x p}. Problem II. Given A* is-an-element-of R(n x n), find A is-an-element-of S(E), such that \\A*-A\\ = inf(A is-an-element-of S(E) \\A*-A\\ where S(E) denotes the solution set of Problem I. The necessary and sufficient conditions for the solvability of Problem I, the expression of the general solution of Problem I and the solution of Problem II are given for two cases. For the general case, the equivalent form of conditions for the solvability of Problem I is given.

对称五对角矩阵加箭型矩阵的广义逆谱问题

研究了一类对称五对角矩阵加箭型矩阵的广义逆谱问题,将该矩阵所有主子阵的极端特征值作为其特征数据,利用几何学上的圆锥曲线、五对角矩阵及箭型矩阵的相关性质,重构此类特殊箭状矩阵。最后给出了该问题解的充分条件以及问题构造的算法与实例,验证了结果的准确性。

两类非对称双箭型矩阵的广义逆谱问题

针对两类非对称双箭型矩阵的广义逆谱问题,本文先将两类矩阵的两组特征对作为其特征数据,然后利用矩阵元素间具有的函数关系、线性关系及箭型矩阵的相关性质,将两类矩阵的逆谱问题转换为求解线性方程组的问题,进而实现了两类矩阵的重构。本文给出了该问题有唯一解的充分必要条件以及问题构造的算法,并通过相应数值实例验证了所得结果。

关于“Key agreement scheme based on generalised inverses of matrices”的一点补充

本文指出E.Dawson和Chuan-Kuan Wu在1997年给出的密钥协定方案须取m≠n,并给出一种遵守任何情况下(包含m=n情况)的修正方案.

ANALYSIS AND SIMULATION OF CONTROL MIXER CONCEPT FOR A CONTROL RECONFIGURABLE AIRCRAFT

In today's aircraft,the hardware redundancy is driven by the critical surfaces resulting in single point-failures.Reconfiguration technology remoVes the single surface criticality by employing control surfaces with aerodynamic redundancy.This paper studies a control reconfiguration scheme based on Control Mixer Concept.A technique for the design of a control mixer for an aircraft with damaged surfaces/actuators using the pseudo-inverse is developed and applied.This paper discusses its applications and limitations based on linear analysis and computer simulation.