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A square matrix is said to be diagonalizable if it is similar to a diagonal matrix. In this paper, we discuss diagonability of matrices over commutative semirings and give an equivalent condition for an idempotent matrix over a commutative semiring to be diagonalizable. Also, we obtain an equivalent description for a matrix over a multiplicatively cancellative and commutative semiring to be diagonalizable. [ABSTRACT FROM AUTHOR]

Fillmore's Theorem says that if A is a nonscalar matrix of order n over a field F and γ 1 , γ 2 , ... , γ n ∈ F are such that γ 1 + γ 2 + ⋯ + γ n = t r A , then there is a matrix B of order n over F similar to A with diagonal (γ 1 , γ 2 , ... , γ n). Borobia extended Fillmore's Theorem to the matrices over the ring Z of integers. In this paper, we will extend Fillmore's Theorem to the matrices over a general factorial ring. [ABSTRACT FROM AUTHOR]

A square matrix over a semiring is called strongly invertible if all of its leading principal submatrices are invertible. In this paper, the strongly invertible matrices over a semiring are discussed and an equivalent condition for a square matrix over a semiring to be strongly invertible is given. Also, some equivalent descriptions are obtained for a semiring over which the product of any two strongly invertible matrices with the same size is strongly invertible. Some of the results obtained in this paper generalize and develop previous results for matrices over the field of complex numbers and matrices over commutative rings. [ABSTRACT FROM AUTHOR]

In this paper, Dedekind’s theorem on the linear independence of homomorphisms of commutative semirings is studied. Furthermore, this theorem is extended to the case of linear independence of compositions of homomorphisms and powers of a derivation. The results obtained in this paper generalize and develop previous results for fields. [ABSTRACT FROM PUBLISHER]

In this paper, the inner products on the semimodules over a commutative semiring are investigated. Some characterizations for orthogonal sets and standard orthogonal basis in the semimodules are given. In particular, an equivalent description is obtained for a semifieldSsatisfying the property that every standard orthogonal set in a finitely generated semimoduleoverScan be extended to a standard orthogonal basis for. Also, the adjoint homomorphisms of the semimodules are discussed and some properties of the adjoint homomorphisms are obtained. Partial results obtained in this paper generalize and develop corresponding results for semilinear spaces ofn-dimensional vectors over commutative zerosumfree semirings and for unitary spaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]