Antiderivations on the exterior algebras

A matrix paraphrase of a certain body of facts dealing with real or complex numbers is a translation of these facts into matrix algebra in which the numbers are replaced by matrices. In two recent papers we developed matrix paraphrases of the Gaussian periods and the Klooster-mann sums. In this paper we paraphrase the theory of finite Fourier series and apply these results to Kloostermann matrices.     

On the Inverses of Brownian and Brownian-Like Matrices

In several papers by F. Valvi, sufficient conditions are given for Brownian and Brownian-like matrices to have Hessenberg inverses. We interpret these conditions from the viewpoint of familiar facts related to matrices of small triangular rank. This allows us to formulate more general assertions on the Hessenberg property of the inverse. Moreover, we explicitly find the structure of the inverse of a Brownian matrix under a certain natural “irreducibility” condition. This structure is similar to the well-known structure of the inverse of an irreducible tridiagonal matrix. Furthermore, we show that the parameters defining the inverse of an n X n Brownian matrix can be calculated in O(n) arithmetic operations.     

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

Generalized fuzzy matrices

We give a systematic development of fuzzy matrix theory. Many of our results generalize to matrices over the two element Boolean algebra, over the nonnegative real numbers, over the nonnegative integers, and over the semirings, and we present these generalizations. Our first main result is that while spaces of fuzzy vectors do not have a unique basis in general they have a unique standard basis, and the cardinality of any two bases are equal. Thus concepts of row and column basis, row and column rank can be defined for fuzzy matrices. Then we study Green's equivalence classes of fuzzy matrices. New we give criteria for a fuzzy matrix to be regular and prove that the row and column rank of any regular fuzzy matrix are equal. Various inverses are also studied. In the next section, we obtain bounds for the index and period of a fuzzy matrix.     

On inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners

In this paper, we derive explicit determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners of Type I. The Mersenne numbers play an important role in these explicit formulas derived. Our main approaches include clever uses of the Schur complement and matrix decomposition with the Sherman-Morrison-Woodbury formula. Besides, the properties of Type II matrix can be also obtained, which benefits from the relation between Type I and II matrices. Lastly, we give three algorithms for these basic quantities and analyze them to illustrate our theoretical results.     

分支数≤7时S^2NS有向图的禁用子图刻划

实方阵A称为强符号非异阵（S~2NS阵）,若任一与A符号模式相同的矩阵非异且其逆的符号模式也一致。若一个有向图是某一S~2NS阵对应赋号有向图的基础有向图,称为S~2NS有向图。本文用禁用子图形式给出了分支数≤7时有向图成为S~2NS有向图的刻划,同时部分地解决了[2]和[4]中提出的问题。     

Identities via Bell matrix and Fibonacci matrix

In this paper, we study the relations between the Bell matrix and the Fibonacci matrix, which provide a unified approach to some lower triangular matrices, such as the Stirling matrices of both kinds, the Lah matrix, and the generalized Pascal matrix. To make the results more general, the discussion is also extended to the generalized Fibonacci numbers and the corresponding matrix. Moreover, based on the matrix representations, various identities are derived.     

Full spark frames and totally positive matrices

In this paper we establish a connection between full spark frames and totally nonsingular matrices. Then we provide a method for constructing infinite totally positive matrices which make up a subclass of the class of totally nonsingular matrices. Using this method we then construct a family of infinite totally positive matrices parameterized by non-negative numbers which contains, as the simplest case, the infinite Pascal matrix. The paper ends with some examples and comments on full spark frames.     

Summation of bounded sequences by regular positive matrices

In terms of elements of regular positive discrete and semicontinuous matrices sufficient conditions are obtained that these matrices do not sum a given real,divergent, bounded sequence and, in particular, that the core of a given sequence coincides with the core of the sequence (function) transformed with the aid of a matrix. These conditions permit certain facts, some well-known and some new, on the theory of inefficient matrices to be obtained.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 179–188, February, 1973.     

由谱数据构造实双反对称矩阵

讨论实完全反对称矩阵的一个特秆值反问题．研究了实完全反对称矩阵的一些特征性质，构造一个实反对称矩阵使其各阶顺序主子矩阵具有指定的特征值．证明了：给定满足一定分隔条件的两组数，存在一个实完全反对称矩阵，使其各阶中心主子矩阵具有相应的特征值．     

Weighted singular decomposition and weighted pseudoinversion of matrices

For a rectangular real matrix, we obtain a decomposition in weighted singular numbers. On this basis, we obtain a representation of a weighted pseudoinverse matrix in terms of weighted orthogonal matrices and weighted singular numbers.     

The geometric mean and B-loop quasigroups of the convex cone of positive definite matrices

In this paper we consider the convex cone of positive definite matrices as algebraic system equipped with geometric mean and B-loop from the standard matrix polar decomposition. Some algebraic structures of these quasigroups are investigated in the context of matrix theory. In particular, their autotopism groups are completely determined: they are isomorphic to the group of positive real numbers.Received: 28 April 2004     

On the Adjacency-Jacobsthal numbers

In this article, we define the adjacency-Jacobsthal sequence and then we obtain the combinatorial representations and the sums of adjacency-Jacobsthal numbers by the aid of generating function and generating matrix of the adjacency-Jacobsthal sequence. Also, we derive the determinantal and the permanental representations of adjacency-Jacobsthal numbers by using certain matrices which are obtained from generating matrix of adjacency-Jacobsthal numbers. Furthermore, using the roots of characteristic polynomial of the adjacency-Jacobsthal sequence, we produce the Binet formula for adjacency-Jacobsthal numbers. Finally, we give the relationships between adjacency-Jacobsthal numbers and Fibonacci, Pell, and Jacobsthal numbers.     

On the purely algebraic data‐sparse approximation of the inverse and the triangular factors of sparse matrices

The approximation of the inverse and the factors of the LU decomposition of general sparse matrices by hierarchical matrices is investigated. In this first approach, we present and motivate a new matrix partitioning algorithm which is based on the matrix graph by proving logarithmic‐linear complexity of the approximant in the case of bounded condition numbers. In contrast to the usual partitioning, the new algorithm allows to treat general grids if the origin of the sparse matrix is the finite element discretization of differential operators. Numerical examples indicate that the restriction to bounded condition numbers has only technical reasons. Copyright © 2010 John Wiley & Sons, Ltd.     

A generalization of imaginary parts of eigenvalues for matrices: Chain rotation numbers

Rotation numbers and chain rotation numbers may be interpreted as a generalization of the imaginary parts for matrices. In dimension two they measure how the solutions of a linear autonomous differential equation rotate in the phase space, and they reduce to the imaginary parts of the eigenvalues of the system’s matrix. In higher dimensions they measure how a two-frame of vectors rotate under the induced flow in the plane which is spanned by the frame. For their calculation, only special sets in the oriented Grassmann manifold of planes are relevant, and to each of these sets corresponds a compact interval of chain rotation numbers. In this paper we will determine these relevant sets and calculate the corresponding sets of chain rotation numbers.     

Pascal k-elminated functional matrix and eulerian numbers

In this paper we shall first introduce the Pascal k-eliminated functional matrices P_n,k[xyz] and CP_n,k[xyz]. Then, using these matrices we obtain several important combinatorial identities. Finally, using the matrix inversion of P_n,k[xyz] and CP_n,k[xyz], we derive an interesting formula for Eulerian numbers [7]     

Remarks on the sharp partial order and the ordering of squares of matrices

In this note we revisit the sharp partial order introduced by Mitra [S.K. Mitra, On group inverses and the sharp order, Linear Algebra Appl. 92 (1987) 17-37]. We recall some already known facts from certain matrix decompositions and derive new statements, relating our discussion to recent results in the literature concerned with partial orders between matrices and their squares.     

A Note on Generic Fiedler Matrices

In this paper, we first show that a generic m×n Fiedler matrix may have 2m-n-1 kinds of factorizations which are very complicated when m is much larger than n. In this work, two special cases are examined, one is an m×n Fiedler matrix being factored as a product of (m - n) Fiedler matrices, the other is an m×(m - 2) Fiedler matrix's factorization. Then we discuss the relation among the numbers of parameters of three generic m×n, n×p and m×p Fiedler matrices, and obtain some useful results.     

Generalized Leibniz functional matrices and factorizations of some well-known matrices

In this paper, we introduce the generalized Leibniz functional matrices and study some algebraic properties of such matrices. To demonstrate applications of these properties, we derive several novel factorization forms of some well-known matrices, such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix. In addition, by applying factorizations of the generalized Leibniz functional matrices, we redevelop the known results of factorizations of Stirling matrices of the first and second kind and the generalized Pascal matrix.