G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

Existence theorems and necessary conditions for a general formulation of the minimum effort problem

The basic problem considered may be described briefly as follows. LetX,Y, andZ be normed linear spaces,T:D(T)→Y,S:D(S)→Z linear operators withD(T) \( \subseteq\) X andD(S) \( \subseteq\) X,Ω \( \subseteq\) X a convex set containing the zero elementθ, andJ a real-valued convex function defined onX×Y such that

1. J(x,y)?-0 for (x,y)teX×Y,
2. J(θ,θ)=0,
3. J(x,y)→+∞, as (∥x∥²+∥y∥²)^1/2→+∞.

Givenζ∈Y andη∈S[core _T Ω∩;D(S)], find an elementx=x ₀ which minimizesJ(x,ζ?Tx) on the set {x∈[Ω∩;D(S)∩;D(T)]:Sx=η}. The abovementioned problem, together with certain special cases, is analyzed using the classical techniques of functional analysis. Existence problems are considered for a certain class of closed linear operators. In particular, existence of an optimal solution is determined by evaluating a generalized Minkowski functional at the point (ζ,η) inY×Z. A necessary condition is presented for special cases, and corresponding characterizations of optimal solutions are made in terms of the adjoint operators. These results are applicable to linear minimum effort problems, constrained variational problems, optimal control of distributive systems, and certain ill-posed variational problems.     

The centralizer of a semi-linear transformation

Let V be a finite-dimensional vector space over a division ring D, where D is finite-dimensional over its center F. Suppose T is a semi-linear transformation on V with associated automorphism σ of D. The centralizer of T is the ring C(T) of all linear transformations on V which commute with T. If σ^r is the identity on D for some r ? 1 and no smaller positive power of σ is an inner automorphism, then the center of C(T) is computed to be polynomials in T^r with coefficients from F₀, where F₀ is the subfield of F left elementwise fixed by σ. A matrix version of this theorem is also given.     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

On strong Z-matrices

A strong Z-matrix is a Z-matrix with negative off-diagonal entries and nonnegative diagonal entries. In this article, for a nonsingular strong Z-matrix, we indicate a way to ascertain whether such a matrix has the Lipschitzian property or not.     

Characterization of Hermitian and skew-Hermitian maps between matrix algebras

Let D be a division ring with an involution J such that D is finite-dimensional over its center Z and char D≠2. Let T:M_m(D)→M_n(D) be a Z-linear map between matrix rings over D. We show that T satisfies [T(X)]¹=T(X¹) if and only if T(X)=∑±A¹_kXA_k. Similarly, T satisfies [T(X)]¹ = ? T(X¹) if and only if T(X = ∑(A¹_kXB_k ? B¹_kXA_k). The first of these results generalizes and extends a theorem of R.D. Hill [2] on Hermitian-preserving transformations.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

A trace inequality for M-matrices and the symmetrizability of a real matrix by a positive diagonal matrix

The question of whether a real matrix is symmetrizable via multiplication by a diagonal matrix with positive diagonal entries is reduced to the corresponding question for M-matrices and related to Hadamard products. In the process, for a nonsingular M-matrix A, it is shown that tr(A^-1A^T) ? n, with equality if and only if A is symmetric, and that the minimum eigenvalue of A^-1 ° A is ? 1 with equality in the irreducible case if and only if A is positive diagonally symmetrizable.     

Classes of general H-matrices

Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M-matrix. H-matrices such that their comparison matrices are nonsingular are well studied in the literature. In this paper, we study characterizations of H-matrices with either singular or nonsingular comparison matrices. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, a classification of the set of general H-matrices is obtained.     

Matrices and set intersections

The incidence matrix of a (υ, k, λ)-design is a (0, 1)-matrix A of order υ that satisfies the matrix equation AA^T=(k?λ)I+λJ, where A^T denotes the transpose of the matrix A, I is the identity matrix of order υ, J is the matrix of 1's of order υ, and υ, k, λ are integers such that 0<λ<k<υ?1. This matrix equation along with various modifications and generalizations has been extensively studied over many years. The theory presents an intriguing joining together of combinatorics, number theory, and matrix theory. We survey a portion of the recent literature. We discuss such varied topics as integral solutions, completion theorems, and λ-designs. We also discuss related topics such as Hadamard matrices and finite projective planes. Throughout the discussion we mention a number of basic problems that remain unsolved.     

An extension of Birkhoff’s theorem with an application to determinants

Let M_n(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈M_n(C), that is, J=J^*,J²=I, let us consider Cⁿ endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cⁿ. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j₁,?,j_n)=I_r⊕-I_n-r. For T=(|t_ik|²)∈M_n(R), the matrices of the form T=(|t_ik|²j_ij_k), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.     

M-matrices leading to semiconvergent splittings

An M-matrix as defined by Ostrowski is a matrix that can be split into A = sI ? B, s > 0, B ? 0 with s ? ρ(B), the spectral radius of B. M-matrices with the property that the powers of T = (1/s)B converge for some s are studied and are characterized here in terms of the nonnegativity of the group generalized inverse of A on the range space of A, extending the well-known property that A^{? 1} ? 0 whenever A is nonsingular.     

A note on generalized Lie derivations of prime rings

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D, K: A-R be additive maps such that F([x, y]) = F(x)_y-yK(x)-T(y)_{x + x}D(y) for all x, yEA. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R) > 3 and also in the case A is a noncentral Lie ideal and deg(R) > 9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.     

Noncommutative interpolation and Poisson transforms

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF ^∞ (resp., noncommutative disc algebraA _n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂⁿ are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF ^∞/J on Hilbert spaces whereJ is anyw ^*-closed, 2-sided ideal ofF ^∞, are obtained and used to construct aw ^*-continuous,F ^∞/J-functional calculus associated to row contractionsT=[T ₁,…,T _n] whenf(T₁, …, T_n)=0 for anyf∈J. Other properties of the dual algebraF ^∞/J are considered. The second author was partially supported by NSF DMS-9531954.     

Conley's spectral sequence via the sweeping algorithm

In this article we consider a spectral sequence (Er,dr) associated to a filtered Morse-Conley chain complex (C,Δ), where Δ is a connection matrix. The underlying motivation is to understand connection matrices under continuation. We show how the spectral sequence is completely determined by a family of connection matrices. This family is obtained by a sweeping algorithm for Δ over fields F as well as over Z. This algorithm constructs a sequence of similar matrices Δ⁰=Δ,Δ¹,… , where each matrix is related to the others via a change-of-basis matrix. Each matrix Δr over F (resp., over Z) determines the vector space (resp., Z-module) Er and the differential dr. We also prove the integrality of the final matrix ΔR produced by the sweeping algorithm over Z which is quite surprising, mainly because the intermediate matrices in the process may not have this property. Several other properties of the change-of-basis matrices as well as the intermediate matrices Δr are obtained.     

Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems

In this paper we study linear differential systems (1) $\text{x′ =}\text{A}\text{?}\text{(θ + ωt)x,}\text{where}\text{A}\text{?}\text{(θ)}\text{is an}\text{(n × n)}$ matrix-valued function defined on the k-torus T^k and (θ, t) → θ + ωt is a given irrational twist flow on T^k. First, we show that if A ? C^N(T^k), where N ? {0, 1, 2,…; ∞; ω}, then the spectral subbundles are of class C^N on T^k. Next we assume that à is sufficiently smooth on T^k and ω satisfies a suitable “small divisors” inequality. We show that if (1) satisfies the “full spectrum” assumption, then there is a quasi-periodic linear change of variables x = P(t)y that transforms (1) to a constant coefficient system y′ =By. Finally, we study the case where the matrix $\text{A}\text{?}\text{(θ + ωt)}$ in (1) is the Jacobian matrix of a nonlinear vector field $\text{?(x)}$ evaluated along a quasi-periodic solution x = φ(t) of (2) $\text{x′ = ?(x)}$. We give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system (z, ?) defined in the vicinity of $\text{Ω = H(φ)}$, the hull of φ, so that the linearized system (1) can be represented in the form z′ = Dz, ?′ = ω, where D is a constant matrix. Our results represent substantial improvements over known methods because we do not require that à be “close to” a constant coefficient system.     

Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions

A function f(x) defined on = ₁ × ₂ × … × _n where each _i is totally ordered satisfying f(x y) f(x y) ≥ f(x) f(y), where the lattice operations and refer to the usual ordering on , is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies −DΣ⁻¹D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

The regular graph of a commutative ring

Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = R\Z(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2 ⁿ , where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices.