The Cayley-Hamilton theorem (CHT) is a classic result in linear algebra over fields which states that a matrix satisfies its own characteristic polynomial. CHT has been extended from fields to commutative semirings by Rutherford in 1964. However, to the best of our knowledge, no result is known for noncommutative semirings. This is a serious limitation, as the class of regular languages, with f...

‎A rank equality is established for the sum of finitely many tripotent matrices via elementary block matrix operations‎. ‎Moreover‎, ‎by using this equality and Theorems 8 and 10 in [Chen M‎. ‎and et al‎. ‎On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications‎, ‎The Scientific World Journal 2014 (2014)‎, ‎Article ID 702413‎, ‎7 page...

This paper deals with the Hermitian H(A) and skew-Hermitian part S(A) of a complex matrix A. We characterize all complex matrices A such that H(A), respectively S(A), is a potent matrix. Two approaches are used: characterizations of idempotent and tripotent Hermitian matrices of the form [ X Y ∗ Y 0 ] , and a singular value decomposition of A. In addition, a relation between the potency of H(A)...

In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero-diagonal real matrix A is characterized by A being a Kleene star. We give applications to alcoved polytopes, using normal idempotent matrices (which form a subclass of Kleene stars). For a normal matrix we define a norm and show that this is the radius o...

Abstract Extending an approach considered by Radjawi and Rosenthal (2002), we investigate the set of square matrices whose square equals a linear combination of the matrix itself and an idempotent matrix. Special attention is paid to the Moore-Penrose and group inverse of matrices belonging to this set. References: Radjavi, H. and P. Rosenthal (2002). On commutators of idempotents. Linear and M...

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results their pairwise comparisons according two criteria, subject constraints on ratings. formulate as bi-objective optimization constrained matrix approximation in Chebyshev sense logarithmic scale. The is approximate comparison matrices for each criterion simultaneously by common consistent unit rank,...

This paper deals with the Hermitian H(A) and skew-Hermitian part S(A) of a complex matrix A. We characterize all complex matrices A such that H(A), respectively S(A), is a potent matrix. Two approaches are used: characterizations of idempotent and tripotent Hermitian matrices of the form [ X Y ∗ Y 0 ] , and a singular value decomposition of A. In addition, a relation between the potency of H(A)...