Transitive reduction of a rectangular boolean matrix

We consider a problem of reducing rectangular boolean matrices. Some operations of reduction are defined and some interesting properties are obtained. The results are useful for models of information retrieval systems. Especially they are useful for simplification of document-term matrices.     

On a matrix nilpotent filter

We define two families of homogeneous ideals of the algebra of polynomials generated by power entries of the general matrix and its operator invariants. We study the combinatorial characteristics of these ideals and, in greater detail, the case of second order.     

Some nilpotent and locally nilpotent matrix groups

Say a division ring D is special if for every finite subset X of D there is a homomorphism of the subring of D generated by X into a division ring of finite Schur index a power of its positive characteristic. (D is not assumed to have positive characteristic.) We make a detailed study of nilpotent and locally nilpotent matrix groups over special division rings.

This has been done previously for a number of ‘special’ division rings arising from group algebras and Lie algebras, particularly by A.I. Lichtman. The present paper therefore presents single proofs of all these results. It also covers many division rings not considered before and produces some new results for those that have been considered before.

In view of the definition of ‘special’ it is not surprising that the proofs depend on a detailed analysis of the finite-dimensional case.     

On the dimension of the algebra generated by a boolean matrix

We investigate the subspace of the space of all n × n Boolean (0,1)-matrices, spanned by the powers of an arbitrary matrix. We estimate the maximum dimension of such spaces as a function of n and show that their bases consists of consecutive integer powers of the matrix, starting at I. We also determine the maximum dimension of the space spanned by the powers of as symmetric matrix and characterise the matrices achieving that maximum.     

On the dimension of the algebra generated by a boolean matrix

We investigate the subspace of the space of all n × n Boolean (0,1)-matrices, spanned by the powers of an arbitrary matrix. We estimate the maximum dimension of such spaces as a function of n and show that their bases consists of consecutive integer powers of the matrix, starting at I. We also determine the maximum dimension of the space spanned by the powers of as symmetric matrix and characterise the matrices achieving that maximum.     

Numerical range of a nilpotent matrix and related central force

In this paper, we introduce a new class of upper triangular nilpotent matrices, etermine the eigenvalues of the Hermitian part of rotating the type of nilpotent matrices and relate the numerical range of that typical type of matrices to the orbit of a point mass under a central force.     

Picturing the lattice of invariant subspaces of a nilpotent complex matrix

Let Q be a nilpotent transformation acting on a finite-dimensional complex vector space. A method is given by which a diagrammatic representation of Lat Q, the lattice of invariant subspaces of Q, can be obtained. Basically the method consists of adding to the Hasse diagram of Hyperlat Q the finite lattice of hyperinvariant subspaces of Q in a specified way. Some applications are given.     

When is the numerical range of a nilpotent matrix circular?

The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces tr M^∗M² and tr M^∗M³ are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5×5 nilpotent matrix has circular numerical range are thoroughly discussed.     

Factorization of a singular matrix into nilpotent matrices with prescribed ranks

We give necessary and sufficient conditions for a singular matrix over an arbitrary field to be factorized into a product of two nilpotent matrices with prescribed ranks.     

On nilpotent subsemigroups of the matrix semigroup over an antiring

In this paper, nilpotent subsemigroups in the matrix semigroup over a commutative antiring are discussed. Some basic properties and characterizations for the nilpotent subsemigroups are given, and some equivalent conditions for the matrix semigroup over a commutative antiring to have a maximal nilpotent subsemigroup are obtained. Also, the maximal nilpotent subsemigroups in the matrix semigroup are described.     

Full reduction of a general square matrix

A technique is given for permuting the rows and columns of a general square matrixM into a fully reduced matrixC. The corresponding algorithm is described, its validity is proved and it is compared with Harary's algorithm [5].     

Embeddings on a boolean cube

In this paper, we characterize a class of graphs which can be embedded on a boolean cube. Some of the graphs in this class are identified with the well known graphs such asmulti-dimensional mesh of trees, tree of meshes, etc. We suggest (i) an embedding of anr-dimensional mesh of trees ofn ^r (r+1)–rn ^r–1 nodes on a boolean cube of (2n) ^r nodes, and (ii) an embedding of a tree of meshes with 2n ² logn+n ² nodes on a boolean cube withn ² exp₂ (log (2 logn+1)]) nodes.