Set inertia-preserving matrices

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.     

Set inertia-preserving matrices∗ †

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.     

加强p除环上自共轭矩阵的几个定理

本文将实对称矩阵和复Hermiitian环矩阵,以及更特殊的正定与半正定矩阵的一些较为深入的定理推广到加强p除上矩阵中来.     

Hyponormal and strongly hyponormal matrices in inner product spaces

Complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied. Based on earlier works on normal matrices, the notions of hyponormal and strongly hyponormal matrices are introduced. A full characterization of such matrices is given and it is shown how those matrices are related to different concepts of normal matrices in degenerate inner product spaces. Finally, the existence of invariant semidefinite subspaces for strongly hyponormal matrices is discussed.     

Inversion of polynomial and rational matrices

The inversion of polynomial and rational matrices is considered. For regular matrices, three algorithms for computing the inverse matrix in a factored form are proposed. For singular matrices, algorithms of constructing pseudoinverse matrices are considered. The algorithms of inversion of rational matrices are based on the minimal factorization which reduces the problem to the inversion of polynomial matrices. A class of special polynomial matrices is regarded whose inverse matrices are also polynomial matrices. Inversion algorithms are applied to the solution of systems with polynomial and rational matrices. Bibliography: 3 titles. Translated by V. N. Kublanovskaya. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 97–109.     

广义模糊幂零矩阵

路代数是加法幂等的半环,它包括了布尔代数,模糊代数,分配格及斜坡.因此布尔矩阵,模糊矩阵,格矩阵及斜矩阵都是路代数上的典型矩阵.广义模糊幂零矩阵指的就是路代数上的幂零矩阵.在2010年,Tan研究了路代数上矩阵的幂零性.在Tan的基础上继续讨论了路代数上幂零矩阵的幂零指数.     

Automorphisms of certain groups and semigroups of matrices

Characterizations are given for automorphisms of semigroups of nonnegative matrices including doubly stochastic matrices, row (column) stochastic matrices, positive matrices, and nonnegative monomial matrices. The proofs utilize the structure of the automorphisms of the symmetric group (realized as the group of permutation matrices) and alternating group. Furthermore, for each of the above (semi)groups of matrices, a larger (semi)group of matrices is obtained by relaxing the nonnegativity assumption. Characterizations are also obtained for the automorphisms on the larger (semi)groups and their subgroups (subsemigroups) as well.     

Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices

We define the notion of an orbit matrix with respect to standard weighing matrices, and with respect to types of weighing matrices with entries in a finite field. In the latter case we primarily restrict our attention the fields of order 2, 3 and 4. We construct self-orthogonal and Hermitian self-orthogonal linear codes over finite fields from these types of weighing matrices and their orbit matrices respectively. We demonstrate that this approach applies to several combinatorial structures such as Hadamard matrices and balanced generalized weighing matrices. As a case study we construct self-orthogonal codes from some weighing matrices belonging to some well known infinite families, such as the Paley conference matrices, and weighing matrices constructed from ternary periodic Golay pairs.     

Determinants of matrices over semirings

In this paper, the concept of determinants for the matrices over a commutative semiring is introduced, and a development of determinantal identities is presented. This includes a generalization of the Laplace and Binet–Cauchy Theorems, as well as on adjoint matrices. Also, the determinants and the adjoint matrices over a commutative difference-ordered semiring are discussed and some inequalities for the determinants and for the adjoint matrices are obtained. The main results in this paper generalize the corresponding results for matrices over commutative rings, for fuzzy matrices, for lattice matrices and for incline matrices.     

幂幺矩阵的充要条件

研究幂幺矩阵的充要条件,利用矩阵的秩和齐次线性方程组解空间的维数.将m=2时幂幺矩阵的充要条件推广到一般幂幺矩阵的充要条件,得出了幂幺矩阵可对角化的结果,并将幂幺矩阵的充要条件平行地推广到幂幺线性变换.     

H-MATRICES AND S-DOUBLY DIAGONALLY DOMINANT MATRICES

§1　IntroductionSpecial matrices,especially H-matricesand M-matrices,have very wide applications innumerical calculations,control theory,mathematical physics,optimization techniques andso on.In recenttwo orthree decades,the studies in these matrices are fruitful,and manygraceful equivalentconditions to M-matrices have been proposed.By contrast,though theH-matrices are closely related with M-matrices,researches on H-matrices show that theproblems in H-matrices are more difficultand some res…     

Inclusion relations between certain sets of matrices

In this paper we study inclusion relations between the following four classes of matrices: normal matrices, matrices with equal spectral radius and spectral norm, matrices whose numerical range coincides with the convex polygon spanned by their eigenvalues, and matrices with equal numerical and spectral radii.     

A geometric treatment of generalized inverses and semigroups of nonnegative matrices

The purpose of this paper is to provide a unified treatment from the geometric viewpoint of the following closely related aspects of nonnegative matrices: nonnegative matrices with nonnegative generalized inverses of various kinds; nonnegative rank factorization; regular elements, Green's relations, and maximal subgroups of the semigroups of nonnegative matrices, stochastic matrices, column stochastic matrices, and doubly stochastic matrices.     

A quasiseparable approach to five-diagonal CMV and Fiedler matrices

Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results.Previous work has focused on characterizing (H,1)-quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between (H,1)-quasiseparable matrices and the subclass of (1,1)-quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices.     

REPRINT OF: A quasiseparable approach to five-diagonal CMV and Fiedler matrices

Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results.Previous work has focused on characterizing (H,1)-quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between (H,1)-quasiseparable matrices and the subclass of (1,1)-quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices.     

立体阵的一般结构

本文给出了立体阵的各种表示形式及立体阵乘法的各种定义,推导出其主要性质,说明立体阵的乘积在适当情况下可转化成普通矩阵乘积。然后讨论了立体阵的乘积与矩阵半张量积的关系,并用矩阵半张量积统一了各种立体阵的乘法运算。最后以对策论为例说明它的应用。     

循环矩阵的两种特性的判定

揭示几类矩阵之间的紧密联系.借助于群的子群的判定以及循环布尔矩阵是本原矩阵的判定方法,得到循环模糊矩阵成为幂等矩阵的充要条件,反循环布尔矩阵成为本原矩阵的充要条件.并给出了循环模糊矩阵成为幂等矩阵的判定方法,反循环布尔矩阵成为本原矩阵的判定方法.     

Sums of nilpotent matrices

We study matrices over general rings which are sums of nilpotent matrices. We show that over commutative rings all matrices with nilpotent trace are sums of three nilpotent matrices. We characterize 2-by-2 matrices with integer entries which are sums of two nilpotents via the solvability of a quadratic Diophantine equation. Some exemples in the case of matrices over noncommutative rings are given.     

整环上几类矩阵空间的保持伴随矩阵线性算子

本文刻画了整环上的全矩阵空间、对称矩阵空间和上三角矩阵空间上保持伴随矩阵的线性算子的结构。