体上两个矩阵乘积的群逆

给出了体上两个矩阵乘积的群逆的存在性的一个等价条件及若干充分条件。     

长方对偶矩阵的加权对偶群逆的存在性与表示

本文研究了长方对偶矩阵的加权对偶群逆的存在性与表示问题.利用矩阵的秩和分块表示等给出了长方对偶矩阵的加权对偶群逆存在的若干充分必要条件,并在加权对偶群逆存在的情形下给出了其表达式,推广了对偶群逆的相关结论.通过数值例子说明了加权对偶群逆存在时的计算方法.     

一类分块矩阵群逆的表达式

鉴于分块矩阵的群逆在许多领域都有重要的应用,根据矩阵投影性质和初等分解的方法给出了分块矩阵M=(AX+YB AB D)在一些新的条件下群逆的存在性理论,然后根据群逆存在性的理论给出群逆的具体表达式.最后通过数值例子验证了结果的正确性.     

正交投影算子乘积广义逆的表示及其性质

本文利用正交投影算子分块形式的表示式,给出了两个投影算子P,Q乘积的MoorePenrose逆以及Drazin逆的表示,并利用所得结果给出了P,Q乘积Drazin逆的相关等式和性质.最后得到了投影算子P,Q的Moore-Penrose逆以及Drazin逆反序律之间的等价关系.     

关于算子加权广义逆的反序律

本文研究了Hilbert空间上两个算子乘积的加权广义逆的反序律.利用算子的分块矩阵表示,获得了两个算子加权广义逆反序律成立的充要条件,所获结果推广了孙文瑜,魏益民和Djordjevic Dragan S.的相关结果.     

两个幂等矩阵的组合的群逆

本文研究了当P与Q是两个复数域上的n阶幂等矩阵且满足PQP=PQ时,组合aP+bQ+cP Q+dQP+eQP Q的群逆问题,利用矩阵的分块及群逆的性质,证明了它是群逆阵,并且给出了其群逆的表达式,其中ab=0,a,b,c,d,e为复数.     

四元数体上矩阵方程的自共轭解

利用矩阵的M-P逆和矩阵分块,给出了四元数体上矩阵方程XB=D在子空间上有自共轭解的充要条件以及解的一般形式,并由此给出了矩阵方程AXB=D有自共轭解的充要条件和解的一般形式.     

正定矩阵的Khatri-Rao乘积的块Schur补的逆的一些偏序

给出了分块矩阵的块Schur补的定义，得到一些正定矩阵的Khatri-Rao乘积的块Schur补的逆的偏序,推广了正定矩阵的Hadamare乘积的相应结果。     

关于长方矩阵的加权群逆的存在性

本文讨论长方矩阵的加权群逆．分别利用减逆和泛分解,给出了长方矩阵的加权群逆存在的几个充要条件以及加权群逆的计算公式．     

群逆和Drazin逆的一种新的表达式及其应用

本文研究了群逆的存在条件及群逆、Drazin逆的表示与计算.利用行列式表示方法,得到了群逆存在的充要条件,给出了群逆的与原矩阵最大非奇异子阵有关的表达式.并推广到Drazin逆.为群逆和Drazin逆的计算提供了一类新的算法.     

除环上左线性方程组的反问题

推广并改进了实数域上线性方程组的反问题及其一系列结果，解决了除环上左线性方程组更具广泛性的一类反问题，给出了此类反问题有（斜）自共轭解及（半）正定自共轭解的充要条件及其解集结构．     

On regularity of block triangular fuzzy matrices

Necessary and sufficient conditions are given for the regularity of block triangular fuzzy matrices. This leads to characterization of idempotency of a class of triangular Toeplitz matrices. As an application, the existence of group inverse of a block triangular fuzzy matrix is discussed. Equivalent conditions for a regular block triangular fuzzy matrix to be expressed as a sum of regular block fuzzy matrices is derived. Further, fuzzy relational equations consistency is studied.     

Constructions of optimal LCD codes over large finite fields

In this paper¹, we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.     

Block toeplitz products of block toeplitz matrices

Necessary and sufficient conditions for the product of two block Toeplitz matrices to be block Toeplitz are obtained. In the special case of two Toeplitz matrices, the conditions simplify considerably and, when combined with known necessary and sufficient conditions for a nonsingular Toeplitz matrix to have a Toeplitz inverse, provide a simple characterization of the additional matrix structure required by a subclass of Toeplitz matrices in order for it to be closed with respect to both inversion and multiplication.     

体上右线性方程组的反问题

设Ｆ，Ｋ，Ω分别表示一个任意的体、一个具有对合反自同构的体和一个实四元数体，Ｆ^ｎ表示Ｆ上的ｎ维右向量空间．本文推广和改进了实线性方程组的反问题及一系列结果，解决了Ｆ上右线性方程组更具一般性的反问题（简称ＩＰＳ）：给定ｂ∈Ｆ^s和α_ｉ∈Ｆ^ｎ（ｉ＝１，…，ｍ≤ｎ）满足ｒａｎｋ［α₁，…，α_m］＝ｍ，求所有的ｓ×ｎ矩阵Ａ使Ａα_ｉ＝ｂ（ｉ＝１，…，ｍ）．当ｓ＝ｎ时     

环上矩阵广义Moore-Penrose逆的存在性

讨论带有对合反自同构*有单位元的结合环R上矩阵的广义Moore-Penrose 逆,给出了环R上矩阵的广义Moore-Penrose逆存在的几个充要条件．特别,得到了环 R上矩阵A的关于M和N的广义Moore-Penrose逆存在的充要条件是A有分解A= GDH,其中D2=D,(MD)*=MD,(GD)*MGD+M(I-D)和DHN-1(DH)*+ (I-D)M-1均可逆．     

Decomposition of Matrices into Involutory Matrices and Symmetric Matrices

Let \Omega be a field, and let F denote the Frobenius matrix: $[F = \left( {\begin{array}{*{20}{c}} 0&{ - {\alpha _n}}\{{E_{n - 1}}}&\alpha \end{array}} \right)\]$ where \alpha is an n-1 dimentional vector over Q, and E_n- 1 is identity matrix over \Omega. Theorem 1. There hold two elementary decompositions of Frobenius matrix: (i) F=SJB, where S, J are two symmetric matrices, and B is an involutory matrix; (ii) F=CQD, where O is an involutory matrix, Q is an orthogonal matrix over \Omega, and D is a diagonal matrix. We use the decomposition (i) to deduce the following two theorems: Theorem 2. Every square matrix over \Omega is a product of twe symmetric matrices and one involutory matrix. Theorem 3. Every square matrix over \Omega is a product of not more than four symmetric matrices. By using the decomposition (ii), we easily verify the following Theorem 4(Wonenburger-Djokovic') . The necessary and sufficient condition that a square matrix A may be decomposed as a product of two involutory matrices is that A is nonsingular and similar to its inverse A^-1 over Q (See [2, 3]). We also use the decomosition (ii) to obtain Theorem 5. Every unimodular matrix is similar to the matrix CQB, where C, B are two involutory matrices, and Q is an orthogonal matrix over Q. As a consequence of Theorem 5. we deduce immediately the following Theorem 6 (Gustafson-Halmos-Radjavi). Every unimodular matrix may be decomposed as a product of not more than four involutory matrices (See [1] ). Finally, we use the decomposition (ii) to derive the following Thoerem 7. If the unimodular matrix A possesses one invariant factor which is not constant polynomial, or the determinant of the unimodular matrix A is I and A possesses two invariant factors with the same degree (>0), then A may be decomposed as a product of three involutory matrices. All of the proofs of the above theorems are constructive.     

环上矩阵的广义Moore-Penrose逆

本文研究环上矩阵的广义Moore-Penros逆,利用矩阵行空间与列空间的包含关系,给出其存在的充要条件及表达式．推广了以往文献的相应结果。