Primes In The Doubly Stochastic Matrices And In The Doubly Stochastic Circulants

The problem of matrix factorizations are very important in matrix theory and matrix computations. If we know a kind decomposition of a matrix, then we will know more about the matrix. It is beneficial to our analyses and computations. On the other hand, matrix factorizations are usefull in reality. For example, factorizations of the nonnegative matrices are interest to signal processing, combinatorial optimization, complexty theory, probability, demography and economics etc.In this paper, primes in the nonnegative matrices are explored. On one hand, it arises in decompositions in the nonegative matrices, on the other hand, it is usefull in reality. For example, the stochastic realization problem for finite-valued process, the realization problem for the hidden Morkov model, the realization problem for a finite stochastic system, and the realization problem for a positive linear system(in which inputs, states, and outputs take positive values) etc. in control and system theory(the main question for these realization problems is the characterization of min-imality for these systems. The question reduces to a problem of positive linear algebra, i.e., the classification problem of primes in the nonnegative matrices).In this paper we mainly investigate primes in the doubly stochastic matrices, and in the doubly stochastic circulants. Since any n×n doubly stochastic circulant matrix has a unique representation as a polynomial of degree n-1 in a n×n shift operator, and any n×n doubly stochastic matrix has a representation as a convex sum of permutations, the classification problem of primes in the doubly stochastic circulants can be reduced to the solution of an equation over a doubly stochastic circulant matrix, and the classification problem of primes in the doubly stochastic matrices can be reduced to solvability of an equation over a doubly stochastic latin square.In chapter 1, we give a brief introduction to the importance of inves- tigating primes in the nonnegative matrices in both theory and practice, and to the current research situation on primes. Moreover, some definitions and their relatin on primes in several classes of the nonnegative matrices and in several classes of the semirings which relate to this paper are given. At final, some results on Hurwitz polynomials which are of interest to this paper are given.In chapter 2, we mainly investigate the methods on how to distinguish whether a matrix whose corresponding vector is of consecutive positive components is a prime in the doubly stochastic circulants, and a problem and a conjecture which were posed by G.Picci etc. in [47], that isProblem A let A = circ(a)∈DSC₊^n×n,4 < n(a) < n, and a =ω_n^k(a₁,…,a_n(a),0,…0)^t∈S-+ⁿ, k∈N_n-1 (i.e., the positive components of the vector a are consecutive). how to distinguish whether A is a prime in the doubly stochastic circulants.Conjecture A Let n∈Z₊,n≥6, A = circ(a)∈DSC₊^n×n,n(a) = n - 1. Then A is not a prime in the doubly stochastic circulants.When n(a) = 5, Problem A and Conjecture A have been solved, we also give a sufficient condition of Conjecture A.In chapter 3, the methods on how to distinguish whether a matrix whose corresponding vector is of positive components which are not all consecutive is a prime in the doubly stochastic circulants are investigated, and it is completely solved how to distinguish whether a matrix whose corresponding vector is of. order 3 or 4 and of positive components which are not all consecutive is a prime in the doubly stochastic circulants.In chapter 4, the methods on how to distinguish whether a matrix is a prime in the doubly stochastic matrices are investigated, and the following problem posed by G.Picci etc. in [47] is solved:Problem B LetA∈DS₊^n×n(n≥3) has a unique representation as a convex sum of permutations, that is A = S₊^n! n(a) = 3. Then A is a prime in the doubly stochastic matrices iff there do not exist b, c∈S₊^n!, n(b),n(c)≥2, such thata = L_m(b)c.Where L_m : S₊^n!→R₊^n!×n! be the latin square induced by multiplication of permutations (see Definition 4.2.3). But there exist b,c∈S₊^n!,n(b),n(c)≥2, such thata = L_m(b)ciff the following conditions both holds:WhereThe solvability of the index equation is characterized in [47]. The problem here is: how to characterize the solvability of the corresponding induced latin square.