New Infinite Classes of Orthogonal Designs

In this paper we use some results on weighing matrices W (2 n ,9) constructed using two circulants to obtain infinite classes of orthogonal designs. We also present a method that is used to construct two directed sequences of length n and specific type. We apply this method to W (2 n , 9) and we obtain many new pairs of directed sequences of length n and type $(9 \cdot 2^m,9 \cdot 2^m), \ m=0,1,2,3.$ Using these directed sequences we can obtain many new infinite classes of weighing matrices W (2 n , k ), and orthogonal designs OD (2 n ; a,b) constructed from two circulants.     

Some perturbation results for the Linear Complementarity Problem

It has been shown previously that the Linear Complementarity Problem is stable when the defining matrix is positive semidefinite and when (locally) the set of solutions is nonempty and bounded. We enlarge the class of matrices for which this is true and also demonstrate how the boundedness condition leads to other stability type questions.     

Similarity Classes of 3 × 3 Matrices Over a Local Principal Ideal Ring

In this article similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.     

Linear Operators that Preserve Commuting Pairs of Nonnegative Real Matrices

A pair of n×n matrices (A, B) is called a commuting pair if AB=BA. We characterize the linear operators that preserve the set of commuting pairs of matrices over a subsemiring of nonnegative real numbers.     

On Some Properties of Convex Matrix Sets Characterized by P -Matrices and Block P -Matrices

In this paper, we consider convex sets of real matrices and establish criteria characterizing these sets with respect to certain matrix properties of their elements. In particular, we deal with convex sets of P-matrices, block P-matrices and M-matrices, nonsingular and full rank matrices, as well as stable and Schur stable matrices. Our results are essentially based on the notion of a block P-matrix and extend and generalize some recently published results on this topic.     

An Infinite Family of Hadamard Matrices with Fourth Last Pivot n /2

We show that the equivalence class of Sylvester Hadamard matrices give an infinite family of Hadamard matrices in which the fourth last pivot is n /2. Analytical examples of completely pivoted Hadamard matrices of order n having as fourth last pivot n /2 are given for n = 16 and 32. In each case this distinguished case with the fourth pivot n /2 arose in the equivalence class of the Sylvester Hadamard matrices.     

Additive Decomposition of Real Matrices

The orthogonal orbit ${\cal O}(A)$ of an n × n real matrix A is the set of real matrices of the form $P^t \ AP$ where $P^t P = I_n$ . We show that $A/ \| A\|$ is an affine sum of four orthogonal matrices, and note that $A^t$ can always be written as an affine combination of no more than 2 n m 1 matrices in ${\cal O}(A)$ . This improves some recent results of Zhan, and answers some of his questions. Other related results are also discussed.     

A New Singular Matrix Method for Balancing Chemical Equations and Their Stability

In this work is given a new singular matrix method for balancing new classes of chemical equations which reduce to an n × n matrix. The method offered here is founded by virtue of the solution of a homogeneous matrix equation by using of Drazin pseudoinverse matrix. The method has been tested on many typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method works successfully without any limitations. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are analyzed some necessary and sufficient criteria for stability of chemical equations over stability of their reaction matrices.     

\boldsymbol{\mathcal{Q}}-*-Categories

We consider the theory of categories enriched in an involutive quantaloid : the -*-categories. After giving an introduction to involutive quantaloids and nuclei, we use matrices with entries in to define -*-categories. Then we examine the relations between two kinds of morphisms between them, the functors and the *-maps, to provide a basis to study completeness properties. These results are used to provide a definition of pseudo-presheaves, presheaves and sheaves on involutive quantaloids in order to get a generalization of presheaves and sheaves on sites. Finally a characterization of these sheaves in terms of covers and compatible families is presented.