Clean matrices over commutative rings

A matrix A ∈ M _n (R) is e-clean provided there exists an idempotent E ∈ M _n (R) such that A-E ∈ GL _n (R) and det E = e. We get a general criterion of e-cleanness for the matrix [[a ₁, a ₂,..., a _n +1]]. Under the n-stable range ondition, it is shown that [[a ₁, a ₂,..., a _n +1]] is 0-clean iff (a ₁, a ₂,..., a _n +1) = 1. As an application, we prove that the 0-cleanness and unit-regularity for such n × n matrix over a Dedekind domain coincide for all n ⩾ 3. The analogous for (s, 2) property is also obtained.      

Group actions in a unit-regular ring

Let Rbe a unit-regular ring , let Xbe the set of all nonzero, nonunits of Rand let Gbe the group of all units of R. In this paper, some finiteness properties of Rare investigated by considering group actions of Gon Xas follows:First, in case of half-transitive regualr action if 2 is unit in Ror the number of idempotents in Ris finite, then Ris finite. Secondly, if Gis cyclic and 2 is unit in R, then every orbit under regualr action is a finite set, and so in this case, if Rhas a finite number of idempotents, then Ris finite. Finally, if Fis a field in which 2 is unit and the multiplicative group of all nonzero elenents in Fforms a cyclic group, then Fis finite.     

Decomposition of projective spaces defined by unit-regular jordan pairs

An analogue of the Bialynicki-Birula decomposition of a smooth algebraic variety under an action of the multiplicative group is shown to hold for spaces obtained from Jordan pairs by projective equivalence. The methods are Jordan-theoretic rather than algebraic-geometric and involve unit-regularity and rank functions for Jordan pairs.     

Exchange rings in which all regular elements are one-sided unit-regular

Let R be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in R is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.     

On double-one matrices and double-zero matrices

We investigate two special classes of matrices over GF(2) with certain interesting properties. These properties can be applied to construct nonsingular matrix pairs efficiently and thus provide a solution to the long-key problems of McEIiece's public-key cryptosystem.     

On double-one matrices and double-zero matrices

We investigate two special classes of matrices over GF(2) with certain interesting properties. These properties can be applied to construct nonsingular matrix pairs efficiently and thus provide a solution to the long-key problems of McEIiece's public-key cryptosystem.     

关于径向矩阵与谱矩阵

§1.引言与记号 设A∈C~(s×n),则称 ‖A‖=‖AX‖/‖X‖ 为A的谱模(谱范数),其中‖X‖表示向量X∈C~(n×1)的Euclid范数。即当X=(x_1,…,x_n)~(?)时,‖X‖=(XX)~1/2=sum from i=1 to n(|X_1|~2)~1/2;‖AX‖为向量AX的Euclid范数。 如众周知,我们有如下结论: 引理 1[1]、设A、B∈C~(n×n),则谱模满足范数的三个条件: 1>.恒正性:‖A‖≥0且‖A‖=0 A=0; 2>.齐次性:若α∈C,则‖αA‖=|α|·‖A‖; 3>.三角不等式:‖A+B‖≤‖A‖+‖B‖。     

Inversion of conjugate-Toeplitz matrices and conjugate-Hankel matrices

The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms.     

Characterizations of EP matrices and weighted-EP matrices

A square complex matrix A is said to be EP if A and its conjugate transpose A^∗ have the same range. In this paper, we first collect a group of known characterizations of EP matrix, and give some new characterizations of EP matrices. Then, we define weighted-EP matrix, and present a wealth of characterizations for weighted-EP matrix through various rank formulas for matrices and their generalized inverses.     

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over ${\mathbb{Z}}_4$.     

Functionally commutative matrices and matrices with constant eigenvectors

Several theorems are proved showing when a functionally commutative matrix must have a set of constant eigenvectors. Theorems for the converse implication are also given. Counterexamples to both implications in the general case are shown.     

Mixtures of order matrices and generalized order matrices

This paper deals with matrix representations of linear orders, mixtures of order matrices and the non-integral solutions of the linear systems defining them.     

Embedding matrices in Hadamard matrices

It is shown that if A is any n×n matrix of zeros and ones, and if k is the smallest number not less than n which is the order of an Hadamard matrix, then A is a submatrix of an Hadamard matrix of order k².     

Clean triangulations

A polyhedron on a surface is called a clean triangulation if each face is a triangle and each triangle is a face. LetS _p (resp.N _p ) be the closed orientable (resp. nonorlentable) surface of genusp. If (S) is the smallest possible number of triangles in a clean triangulation ofS, the results are: (N ₁)=20, (S ₁)=24, lim(S _p )p ^–1=4, lim(N _p )p ^–1=2 forp.     

Structural properties of Riordan matrices and extending the matrices

We consider an infinite lower triangular matrix L=[?_n,k]_n,k∈N0 and a sequence Ω=(ω_n)_n∈N0 called the (a,b)-sequence such that every element ?_n+1,k+1 except lying in column 0 can be expressed as

     

Self-adjunctions and matrices

It is shown that the multiplicative monoids of Temperley-Lieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. This self-adjunction underlies the orthogonal group case of Brauer's representation of the Brauer centralizer algebras.     

Symmetric polynomials, Pascal matrices, and Stirling matrices

We consider lower-triangular matrices consisting of symmetric polynomials, and we show how to factorize and invert them. Since binomial coefficients and Stirling numbers can be represented in terms of symmetric polynomials, these results contain factorizations and inverses of Pascal and Stirling matrices as special cases. This work generalizes that of several other authors on Pascal and Stirling matrices.     

Approximating Runge-Kutta matrices by triangular matrices

The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrixA in the Newton process for a triangular matrixT that approximatesA, hereby making the method suitable for parallel implementation. The matrixT is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the diagnoal entries ofT are positive. This means that the linear systems that are to be solved have a non-singular matrix. The research reported in this paper was supported by STW (Dutch Foundation for Technical Sciences).