Incline Matrices Research Articles

Preservers of Completely Positive Matrix Rank for Inclines

A real symmetric matrix A is called completely positive if there exists a nonnegative real \(n\times k\) matrix B such that \(A = BB^{t}\). The smallest value of k for all possible choices of nonnegative matrices B is called the CP-rank of A. We extend the ideas of complete positivity and the CP-rank to matrices whose entries are elements of an incline in a similar way. We classify maps on the set of \(n \times n\) symmetric matrices over certain inclines which strongly preserve CP-rank-1 matrices as well as maps which preserve CP-rank-1 and CP-rank-k. The result suggests that there is a certain standard class of solutions for CP-rank preserver problems on incline matrices.

On invertible matrices over commutative semirings

In this article, the invertible matrices over commutative semirings are studied. Some properties and equivalent descriptions of the invertible matrices are given and the inverse matrix of an invertible matrix is presented by analogues of the classic adjoint matrix. Also, Cramer's rule over a commutative semiring is established. The main results obtained in this article generalize the corresponding results for matrices over commutative rings, for lattice matrices, for incline matrices, for matrices over zerosumfree semirings and for matrices over additively regular semirings.

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.

On Generalized Transitive Matrices

Transitivity of generalized fuzzy matrices over a special type of semiring is considered. The semiring is called incline algebra which generalizes Boolean algebra, fuzzy algebra, and distributive lattice. This paper studies the transitive incline matrices in detail. The transitive closure of an incline matrix is studied, and the convergence for powers of transitive incline matrices is considered. Some properties of compositions of incline matrices are also given, and a new transitive incline matrix is constructed from given incline matrices. Finally, the issue of the canonical form of a transitive incline matrix is discussed. The results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references.

On nilpotency of generalized fuzzy matrices

In this paper, generalized fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra. Some properties and characterizations for nilpotent generalized fuzzy matrices are established, reduction of generalized fuzzy matrices is defined and some properties of reduction of nilpotent generalized fuzzy matrices are obtained. Partial results obtained in this paper generalize the corresponding ones on fuzzy matrices, on lattice matrices and on incline matrices.

An application of incline matrices in dynamic analysis of generalized fuzzy bidirectional associative memories

This paper studies an application of the incline matrix theory in the dynamic analysis of incline-valued fuzzy bidirectional associative memories ( L-FBAMs, for short). It is proved that the strong convergence and the strong stability of an L-FBAM are equivalent to the existence of indices and the convergence in finite steps of the product matrices of connection incline matrices of the L-FBAM, respectively. It is shown that the convergence index, the period of limit-cycles, the stable states and equilibria of a strongly convergent L-FBAM are expressed by the indices, the periods and the standard eigenvectors of the product matrices of connection incline matrices of the L-FBAM, respectively.

On invertible matrices over antirings

An antiring is a semiring which is zerosumfree (i.e., a + b = 0 implies a = b = 0 for any a, b in this semiring). In this paper, the complete description of the invertible matrices over a commutative antiring is given and some necessary and sufficient conditions for a matrix over a commutative antiring to be invertible are obtained. Also, Cramer’s rule over commutative antirings is presented. The main results in this paper generalize and develop the corresponding results for the Boolean matrices, the fuzzy matrices, the lattice matrices and the incline matrices.

On nilpotent incline matrices

Inclines are additively idempotent semirings in which products are less than or equal to either factor. Thus they generalize Boolean algebra, fuzzy algebra and distributive lattice. This paper studies the nilpotent incline matrices in detail. It is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its unique standard eigenvector. The nilpotent matrices over an incline without nilpotent elements are characterized in terms of principal minors, main diagonals, nilpotent indices and adjoint matrices. Also some properties of the reduction of nilpotent matrices over an additively residuated incline without nilpotent elements are established. The results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references.

Invertible incline matrices and Cramer's rule over inclines

Inclines are the additively idempotent semirings in which the products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i.e., the matrices over inclines). In this paper, the complete description of the invertible incline matrices is given. Some necessary and sufficient conditions for an incline matrix to be invertible are studied, Cramer's rule over inclines is presented and the group of invertible incline matrices is investigated. The main results in the present paper generalize and develop the corresponding results in the literatures for the Boolean matrices, the fuzzy matrices and the lattice matrices.

Standard eigenvectors of incline matrices

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. This paper studies the standard eigenvectors of incline matrices (i.e., matrices over inclines) in detail. As the main results, for an incline matrix with index and for a lattice matrix, the structures of all the standard eigenvectors are given and the greatest standard eigenvector is determined. Also it is proved that for an incline matrix with index, the set of all the standard eigenvectors of it coincides with the set of all the standard eigenvectors of its transitive closure. The power convergence of an incline matrix with index is characterized in terms of the sets of standard eigenvectors. The results in the present paper include some previous results for the Boolean matrices, the fuzzy matrices and the lattice matrices among their special cases.

Indices and periods of incline matrices

Inclines are the additively idempotent semirings in which products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i.e., the matrices over inclines). This paper studies the power sequence of incline matrices in detail. A necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated. It is proved that the period of n× n incline matrix with index is a divisor of [ n] and that the set of periods of n× n incline matrices with indices is not bounded from above in the sense of a power of n for all n. An equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established. The stability of the orbits of an incline matrix is considered and a theorem in [Fuzzy Sets and Systems 81 (1996) 227] is pointed out being false. The results in the present paper include some previous results in the literatures which were obtained for the Boolean matrices, the fuzzy matrices and the lattice matrices among their special cases.