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- W26056176 abstract "We consider a special semiring of complex matrices with real and imag- inary parts formed by positive nonsingular matrices taken from sets of pairwise com- muting matrices. We describe an orbital structure of acertain transformation of the mentioned semiring satisfying rules of derivation for sum and product. Preliminaries Matrices with nonnegative or positive elements (2, 4, 10), the systematic study of which dates back to G. Frobenius (1912), belong to important mathematical tools of modelling especially in the theory of stochastic processes (modelling of Marko! processes), in mathematical economy and elsewhere. In the theory of ordinary di!eren tial equations the types of equations solutions of which and systems of dif- ferential equations are expressed by linear combinations of products of elementary functions and exponential functions or vector functions and exponential functions of functional matrices. Considering more details, let us remined the Floquet theo- rem (8) concerning systems of ordinary linear di!eren tial equations with periodical coecien ts, which says that if U is the standard matrix of the Cauchy problem x ! = A(t)x,x ! (0) = ! then there exists a continuously di!eren tiable nonsingular n ! n matrix function P and an n ! n constant matrix R, such that function U(t) can be represented by the ordered pair of matrix functions (P(t),R(t)). Calcula- tions rules considered in this paper are motivated by calculations of these ordered pairs, e. g. (P1(t),R1(t)),(P2(t),R2(t)) can be added whenever R1(t) = R2(t); in the opposite case the sum of corresponding pair is substituted by an ideal element, whereas the usual product of the above pairs is the pair (P1(t)P2(t),R1(t)+R2(t)). In connection with papers (5, 10) and monography (18, Chap. I.) we will analyze the orbital structure of a special di!eren tial operator defined on a certain semiring of complex matrices M = A + iB, where A,B are square nonsingular matrices of order n ! 2 with entirely positive elements (i.e. positive matrices). The considered" @default.
- W26056176 created "2016-06-24" @default.
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- W26056176 date "2004-01-01" @default.
- W26056176 modified "2023-09-28" @default.
- W26056176 title "ORBITAL STRUCTURE OF THE DERIVATION OPERATOR ON A CERTAIN SEMIRING OF NONSINGULAR COMPLEX MATRICES" @default.
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