Automorphisms of the semigroup of invertible matrices with nonnegative elements

In this paper, we prove that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly ordered associative ring on some specially defined subgroup coincides with the composition of an inner automorphism of the semigroup, an order-preserving automorphism of the ring, and a central homothety. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 3–23, 2005.     

Elementary equivalence of semigroups of invertible matrices with nonnegative elements over commutative partially ordered rings

In the paper, we prove that if two semigroups of invertible matrices with nonnegative elements over partially ordered commutative rings are elementarily equivalent, then their dimensions coincide and the corresponding semirings of nonnegative elements are elementarily equivalent.     

Automorphisms of the semigroup of invertible matrices with nonnegative elements over commutative partially ordered rings

In the paper, we prove that if two semigroups of invertible matrices with nonnegative elements over partially ordered commutative rings are elementarily equivalent, then their dimensions coincide and the corresponding semirings of nonnegative elements are elementarily equivalent.     

Automorphisms of the semigroup of nonnegative invertible matrices of order 2 over rings

In this paper, we describe automorphisms of the semigroup G₂(R) of nonnegative invertible matrices if R is a (not necessarily commutative) partially ordered ring without zero divisors with 1/n for some natural number n?>?1.     

On invertible nonnegative Hamiltonian operator matrices

Some new characterizations of nonnegative Hamiltonian operator matrices are given. Several necessary and sufficient conditions for an unbounded nonnegative Hamiltonian operator to be invertible are obtained, so that the main results in the previously published papers are corollaries of the new theorems. Most of all we want to stress the method of proof. It is based on the connections between Pauli operator matrices and nonnegative Hamiltonian matrices.     

Automorphisms of the semigroup of nonnegative invertible matrices of order two over partially ordered commutative rings

In the paper, the automorphisms of the semigroup of nonnegative invertible matrices of order two over a partially ordered commutative ring with 2 invertible are described.     

Endomorphisms of semigroups of invertible nonnegative matrices over ordered rings

Let R be a linearly ordered commutative ring with 1/2 and G _n (R) be the subsemigroup of GL _n (R) consisting of matrices with nonnegative elements. In this paper, we describe endomorphisms of this semigroup for n ≥ 3.     

Groups of quotients of semigroups of invertible nonnegative matrices over skewfields

In this paper we prove that for a linearly ordered skewfield the groups of quitients of the semigroup G _n (D) coincides with the group GL _n (D) for n ≥ 3.     

Strong shift equivalence and shear adjacency of nonnegative square integer matrices

For two square matrices A, B of possibly different sizes with nonnegative integer entries, write A ≈₁ B if A = RS and B = SR for some two nonnegative integer matrices R,S. The transitive closure of this relation is called strong shift equivalence and is important in symbolic dynamics, where it has been shown by R.F. Williams to characterize the isomorphism of two topological Markov chains with transition matrices A and B. One invariant is the characteristic polynomial up to factors of λ. However, no procedure for deciding strong shift equivalence is known, even for 2×2 matrices A, B. In fact, for n × n matrices with n > 2, no nontrivial sufficient condition is known. This paper presents such a sufficient condition: that A and B are in the same component of a directed graph whose vertices are all n × n nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities. For n > 2 this result gives confirmation of strong shift equivalences that previously could not be verified; for n = 2, previous results are strengthened and the structure of the directed graph is determined.     

Tridiagonal matrices with nonnegative entries

In this paper, we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let d denote a nonnegative integer. Let A denote a matrix in R and let denote the roots of the characteristic polynomial of A. We say A is multiplicity-free whenever these roots are mutually distinct and contained in R. In this case E_i will denote the primitive idempotent of A associated with theta_i(0?i?d). We say A is symmetrizable whenever there exists an invertible diagonal matrix Δ∈R such that ΔAΔ^-1 is symmetric. Let Γ(A) denote the directed graph with vertex set {0,1,…,d}, where i→j whenever i≠j and A_ij≠0.Theorem.Assume that each entry ofAis nonnegative. Then the following are equivalent for0≤s,t≤d.

(i)

The graphΓ(A)is a bidirected path with endpointss,t:s↔*↔*↔?↔*↔t.

(ii)

The matrixAis symmetrizable and multiplicity-free. Moreover the(s,t)-entry ofE_itimes(θ_i-θ₀)?(θ_i-θ_i-1)(θ_i-θ_i+1)?(θ_i-θ_d)is independent of i for0≤i≤d, and this common value is nonzero.

Recently Kurihara and Nozaki obtained a theorem that characterizes the Q-polynomial property for symmetric association schemes. We view the above result as a linear algebraic generalization of their theorem.     

The nonnegative rank factorizations of nonnegative matrices

Let A ∈ P^{m × n}_r, the set of all m × n nonnegative matrices having the same rank r. For matrices A in P^{m × n}_n, we introduce the concepts of “A has only trivial nonnegative rank factorizations” and “A can have nontrivial nonnegative rank factorizations.” Correspondingly, the set P^{m × n}_n is divided into two disjoint subsets P₍₁₎ and P₍₂₎ such that P₍₁₎∪P₍₂₎ = P^{m × n}_n. It happens that the concept of “A has only trivial nonnegative rank factorizations” is a generalization of “A is prime in P^{n × n}_n.” We characterize the sets P₍₁₎ and P₍₂₎. Some of our results generalize some theorems in the paper of Daniel J. Richman and Hans Schneider [9].     

Toeplitz matrices with totally nonnegative inverses

It is shown that the inverse of a Toeplitz matrix has only nonnegative minors if the zeros of a certain polynomial are positive or if their arguments are less than π?(k+n), where n is the dimension and k+1 is the bandwidth of the matrix.     

Group inverse for the block matrices with an invertible subblock

Let (A is square) be a square block matrix with an invertible subblock over a skew field K. In this paper, we give the necessary and sufficient conditions for the existence as well as the expressions of the group inverse for M under some conditions.     

On simple factorization of invertible matrices

Let I be an ideal of a ring R. We say that R is a generalized I-stable ring provided that aR+bR=R with a?∈?1+I,b?∈?R implies that there exists a y?∈?R such that a+by?∈?K(R), where K(R)={x?∈?R?∣?? s, t?∈?R such that sxt=1}. Let R be a generalized I-stable ring. Then every A?∈?GL_n (I) is the product of 13n?12 simple matrices. Furthermore, we prove that A is the product of n simple matrices if I has stable rank one. This generalizes the results of Vaserstein and Wheland on rings having stable rank one.     

On powers of nonnegative matrices

We investigate the structure of powers of nonnegative matrices A, and in particular characterize those A for which some power is (essentially) triangular. We also show how the number of irreducible components of A can be determined from its powers.     

Algebraic eigenspaces of nonnegative matrices

The Perron-Frobenius theory for square, irreducible, nonnegative matrices is generalized by studying the structure of the algebraic eigenspace of an arbitrary square nonnegative matrix corresponding to its spectral radius. We give a constructive proof that this subspace is spanned by a set of semipositive vectors and give a combinatorial characterization of both the index of the spectral radius and dimension of the algebraic eigenspace corresponding to the spectral radius. This involves a detailed study of the standard block triangular representation of nonnegative matrices by giving special attention to those blocks on the diagonal having the same spectral radius as the original matrix. We also show that the algebraic eigenspace corresponding to the spectral radius contains a semipositive vector having the largest set of positive coordinates among all vectors in this subspace.     

Intervals of totally nonnegative matrices

Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard ordering are considered. It is proven that if the two bound matrices of such a matrix interval are nonsingular and totally nonnegative (and in addition all their zero minors are identical) then all matrices from this interval are also nonsingular and totally nonnegative (with identical zero minors).     

On nonnegative factorization of matrices

It is shown that a sufficient condition for a nonnegative real symmetric matrix to be completely positive is that the matrix is diagonally dominant.