Ranks of the common solution to six quaternion matrix equations

A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721-732 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.     

四元数体上Hermitian矩阵空间保秩1的加法映射

Let Q be the quaternion division algebra over real field F.Denote by H_n(Q)the set of all n×n hermitian matrices over Q.We characterize the additive maps from H_n(Q) into H_m(Q)that preserve rank-1 matrices when the rank of the image of I_n is equal to n. Let Q_R be the quaternion division algebra over the field of real number R.The additive maps from H_n(Q_R) into H_m(Q_R)that preserve rank-1 matrices are also given.     

A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix

A non-linear structure preserving matrix method for the computation of a structured low rank approximation of the Sylvester resultant matrix S(f,g) of two inexact polynomials f=f(y) and g=g(y) is considered in this paper. It is shown that considerably improved results are obtained when f(y) and g(y) are processed prior to the computation of , and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of , which leads to a linear structure preserving matrix method, or they can be incremented during the computation of , which leads to a non-linear structure preserving matrix method. It is shown that the non-linear method yields a better structured low rank approximation of S(f,g) and that the assignment of f(y) and g(y) is important because may be a good structured low rank approximation of S(f,g), but may be a poor structured low rank approximation of S(g,f) because its numerical rank is not defined. Examples that illustrate the differences between the linear and non-linear structure preserving matrix methods, and the importance of the assignment of f(y) and g(y), are shown.     

An improved non-linear method for the computation of a structured low rank approximation of the Sylvester resultant matrix

This paper reports on improvements to recent work on the computation of a structured low rank approximation of the Sylvester resultant matrix S(f,g)$S(f,g)$ of two inexact polynomials f=f(y)$f=f\left(y\right)$ and g=g(y)$g=g\left(y\right)$. Specifically, it has been shown in previous work that these polynomials must be processed before a structured low rank approximation of S(f,g)$S(f,g)$ is computed. The existing algorithm may still, however, yield a structured low rank approximation of S(f,g)$S(f,g)$, but not a structured low rank approximation of S(g,f)$S(g,f)$, which is unsatisfactory. Moreover, a structured low rank approximation of S(f,g)$S(f,g)$ must be equal to, apart from permutations of its columns, a structured low rank approximation of S(g,f)$S(g,f)$, but the existing algorithm does not guarantee the satisfaction of this condition. This paper addresses these issues by modifying the existing algorithm, such that these deficiencies are overcome. Examples that illustrate these improvements are shown.     

A Deterministic Approach to Linear Programs with Several Additional Multiplicative Constraints

We consider a global optimization problem of minimizing a linear function subject to p linear multiplicative constraints as well as ordinary linear constraints. We show that this problem can reduce to a 2p-dimensional reverse convex program, and present an algorithm for solving the resulting problem. Our algorithm converges to a globally optimal solution and yields an -approximate solution in finite time for any > 0. We also report some results of computational experiment.     

四元数矩阵方程组的η-厄尔米特解

研究了包含η-厄尔米特矩阵的四元数矩阵方程组.用四元数矩阵的秩和广义逆给出了一个包含η-厄尔米特矩阵的四元数矩阵方程组相容的充分必要条件.进一步地,用四元数矩阵的广义逆给出了这个四元数矩阵方程组的通解表达式.     

四元数自共轭矩阵乘积的相似变换

本文证明了下列结果:(i)四元数矩阵A可写成两个自共轭四元数矩阵的乘积A相似于实矩阵A Hermite相似于A~*.(ii)A可写成一个半正定自共轭四元数矩阵与一个自共轭四元数矩阵的乘积A相似于实对角矩阵或者A～diag(D,I_r(×)J_2(O)),其中D是一个实对角矩阵.本文还给出了体上实矩阵AB与BA相似的一个充要条件.     

求解结构矩阵低秩逼近的交替投影方法

结构矩阵低秩逼近在图像压缩、计算机代数和语音编码中有广泛应用.首先给出了几类结构矩阵的投影公式,再利用交替投影方法计算结构矩阵低秩逼近问题.数值试验表明新方法是可行的.     

Inaccurate linear equation system with a restricted-rank error matrix

It is well-known that the solution set of an interval linear equation system is a union of convex polyhedra the number of which increases, in general, exponentially with the problem size. As a consequence, the problem of finding the interval hull of the solution set is NP-hard as J. Rohn and V. Kreinovich proved in [13]. The purpose of this paper is to show that the solution set analysis can be simplified substantially provided the rank of the error matrix is restricted even if the assumption of interval character of data errors is replaced by a more general one. Especially, in the case of a rank-one error matrix we have to look into at most two convex subsets. Besides, a dual approach to describing the solution set is discussed. The original version of this approach was suggested in [7].     

质环上的微商及环的结构

本文证明了如下定理:R为质环,char R≠2,d为R上非零微商,R中无非零诣零元,(?)则R为交换环,或R可嵌入体中.     

When is a matrix unitary or Hermitian plus low rank?

Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing complexity once the matrix has been reduced, for instance, to tridiagonal or Hessenberg form. Recently, fast and reliable eigensolvers dealing with low‐rank perturbations of unitary and Hermitian matrices have been proposed. These structured eigenvalue problems appear naturally when computing roots, via confederate linearizations, of polynomials expressed in, for example, the monomial or Chebyshev basis. Often, however, it is not known beforehand whether or not a matrix can be written as the sum of a Hermitian or unitary matrix plus a low‐rank perturbation. In this paper, we give necessary and sufficient conditions characterizing the class of Hermitian or unitary plus low‐rank matrices. The number of singular values deviating from 1 determines the rank of a perturbation to bring a matrix to unitary form. A similar condition holds for Hermitian matrices; the eigenvalues of the skew‐Hermitian part differing from 0 dictate the rank of the perturbation. We prove that these relations are linked via the Cayley transform. Then, based on these conditions, we identify the closest Hermitian or unitary plus rank k matrix to a given matrix A, in Frobenius and spectral norm, and give a formula for their distance from A. Finally, we present a practical iteration to detect the low‐rank perturbation. Numerical tests prove that this straightforward algorithm is effective.     

两个四元数自共轭矩阵乘积的特征值估计

设A和B都是四元数自共轭半正定矩阵,或者其中之一是正定的,而另一个是自共轭的,本文改进并推广了[2]对乘积AB的特征值估计.     

关于三幂等矩阵的秩特征的研究

本文对已有的关于三幂等矩阵秩的等式作了进一步研究,指出其中有些可以作为判定三幂等矩阵的充要条件,即三幂等矩阵的秩特征等式,本文还证明了有无穷多种三幂等矩阵的秩特征等式形式．     

低秩矩阵优化若干新进展

低秩矩阵优化是一类含有秩极小或秩约束的矩阵优化问题,在统计与机器学习、信号与图像处理、通信与量子计算、系统识别与控制、经济与金融等众多学科领域有着广泛应用,是当前最优化及其相关领域的一个重点研究方向.然而,低秩矩阵优化是一个NP-难的非凸非光滑优化问题,其研究成果并非十分丰富,亟待进一步深入研究.主要从理论和算法两个方面总结和评述若干新结果,同时列出相关的重要文献,奉献给读者.     

实四元数矩阵行列式函数的几点注记

分别对实四元数矩阵理论中出现的几种行列式函数在三个初等变换下的改变性作一些讨论,得出几种满足一定性质的行列式函数的不存在性或唯一存在性。     

Schur定理在四元数体上的推广

In this paper,we show that every matrix over the real quaternion division ring is unitary similar to an upper triangular matrix.