On a decomposition of conditionally positive-semidefinite matrices

A symmetric matrix C is said to be copositive if its associated quadratic form is nonnegative on the positive orthant. Recently it has been shown that a quadratic form x'Qx is positive for all x that satisfy more general linear constraints of the form Ax?0, x≠0 iff Q can be decomposed as a sum Q=A'CA+S, with Cstrictly copositive and S positive definite. However, if x'Qx is merely nonnegative subject to the constraints Ax?0, it does not follow that Q admits such a decomposition with C copositive and S positive semidefinite. In this paper we give a characterization of those matrices A for which such a decomposition is always possible.     

Finite criteria for conditional definiteness of quadratic forms

A real symmetric n × n matrix Q is A-conditionally positivesemidefinite, where A is a given m × n matrix, if x′Qx?0 whenever Ax?0, and is A-conditionally positive definite if strict inequality holds except when x=0. When A is the identity matrix these notions reduce to the well-studied notions of copositivity and strict copositivity respectively. This paper presents finite criteria, involving only the solution of sets of linear equations constructed from the matrices Q,A, for testing both types of conditional definiteness. These criteria generalize known facts about copositive matrices and, when Q is invertible and all row submatrices of A have maximal rank, can be very elegantly stated in terms of Schur complements of the matrix AQ^-1A′.     

On the zeros of derivatives of Bessel functions

In this paper we are interested in the behaviour respect tov of thekth positive zeroc′ _vk of the derivative of the general Bessel functionC _v(x)=J _v(x)cosα?Y _v(x)sinα, 0≤α<π, whereJ _v(x) andY _v(x) indicate the Bessel functions of first and second kind respectively. It is well known that forc′ _vk>∥v∥,c′ _vk increases asv increases. Here we prove several additional properties forc′ _vk. Our main result is thatc′ _vk is concave as a function ofv, whenc′ _vk>∥v∥>0. This implies the concavity ofc′ _vk for everyk=2,3, ?. In the case of the zerosJ′ _vk of _d ^dx J _v(x) we extend this property tok=1 for everyv≥0.     

An efficient Newton's method for optimization under equality constraints

An efficient procedure for optimizing a nonlinear objective functional ?(x) under linear and/or nonlinear equality constraints is given. The linearly constrained, quadratic ?(x) case is shown to have a solution given by the explicit formula x = x_p - N(N′AN)^-1N′(Ax_p + b/2), where ?(x) = a+b′x+x′Ax(x?Rⁿ) is convex, and both x_p?R_n and N [an n×(n-r) matrix]; can be obtained simultaneously from the constraint set, Kx=c (K of rank r<n), by a single Gaussian elimination. The nonlinearly constrained, arbitrary ?(x) case is treated by an interative scheme in which the above formula is used to “project” onto approximate solutions satisfying linear approximations of the constraints. This method does not require the initial guess or the iterated values to be in the feasible region. The resulting algorithm does appear to be efficient.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

A simplex procedure for linear programs with some 0–1 integer constraints

For the problem “maximize cx, subject to Ax?b, x?0 and a single 0–1 integer constraint dy?β, β?0, y_i = 1 when x_i > 0 and zero otherwise”, it is shown that only the feasible extreme points of Ax?b, x?0 need be searched. An adjacent vertex cut technique is developed to exclude, one at a time, the extreme points which are infeasible with respect to dy?β. In the concluding section, we also discuss the simple extension of the procedure to the case of multiple 0–1 constraints.     

Overdetermined linear systems satisfying nonnegativity constraints

Let A be a nonnegative m × n matrix, and let b be a nonnegative vector of dimension m. Also, let S be a subspace of $\text{R}$ⁿ such that if P_S is the orthogonal projector onto S, then P_S ? 0. A necessary condition is given for the matrix A to satisfy the following property: For all b ? 0, if min[boxV]b ? Ax[boxV] is attained at x = x₀, then x₀ ? 0 and x₀ ? S. It is also shown that if a nonnegative matrix A has a nonnegative generalized inverse, then any submatrix of A also possesses a nonnegative generalized inverse.     

Norm properties of C-numerical radii

Given n×n complex matrices A, C, the C-numerical radius of A is the nonnegative quantity

$\text{r}{}_{\mathrm{c}}\text{(A)≡ma{|tr(CU}{}^1\text{AU)|:U unitary}}$

. For C=diag(1,0,…,0) it reduces to the classical numerical radius $\text{r(A)= max{|x}{}^1\text{Ax|:x}{}^1\text{x=1}}$. We show that r_c is a generalized matrix norm if and only if C is nonscalar and trC≠0. Next, we consider an arbitrary generalized matrix norm and characterize all constants v?0 for which vN is multiplicative. A technique to obtain such v is then applied to C-numerical radii with Hermitian C. In particular we find that vr is a matrix norm if and only if v?4.     

Two problems of the theory of quadratic maps

Given quadratic forms q ₁, …, q _k , two questions are studied: Under what conditions does the set of common zeros of these quadratic forms consist of the only point x = 0? When is the maximum of these quadratic forms nonnegative or positive for any x ≠ 0? Criteria for each of these conditions to hold are obtained. These criteria are stated in terms of matrices determining the quadratic forms under consideration.     

A theorem of the alternatives for the equation Ax + B|x| = b

The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax?+?B|x|?=?b has a unique solution for each B with |B|?≤?D and for each b, or the equation Ax?+?B ₀|x|?=?0 has a nontrivial solution for some matrix B ₀ of a very special form, |B ₀|?≤?D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax|?=?λ|x| for some x?≠?0, and we prove that each square real matrix has an absolute eigenvalue.     

A copositive Q-matrix which is notR 0

Jeter and Pye gave an example to show that Pang's conjecture, thatL ₁ ?Q ?R ₀, is false while Seetharama Gowda showed that the conjecture is true for symmetric matrices. It is known thatL ₁-symmetric matrices are copositive matrices. Jeter and Pye as well as Seetharama Gowda raised the following question: Is it trueC ₀ ?Q ?R ₀? In this note we present an example of a copositive Q-matrix which is notR ₀. The example is based on the following elementary proposition: LetA be a square matrix of ordern. SupposeR ₁ =R ₂ whereR _i stands for theith row ofA. Further supposeA ₁₁ andA ₂₂ are Q-matrices whereA _ii stands for the principal submatrix omitting theith row andith column fromA. ThenA is a Q-matrix.     

Optimality conditions for quadratic programming

This paper establishes a set of necessary and sufficient conditions in order that a vectorx be a local minimum point to the general (not necessarily convex) quadratic programming problem:minimizep ^T x + 1/2x ^T Qx, subject to the constraintsHx h.     

Linear combinations of Hermitian and real symmetric matrices

This paper, by purely algebraic and elementary methods, studies useful criteria under which the quadratic forms x′Ax and x′Bx, where A,B are n × n symmetric real matrices and x′=(x₁,x₂, …,x_n)≠(0,0,0,0, …,0), can vanish simultaneously and some real linear combination of A,B can be positive definite. Analogous results for hermitian matrices have also been discussed. We have given sufficient conditions on m real symmetric matrices so that some real linear combination of them can be positive definite.     

On matrix measures and convex Liapunov functions

In this paper, we extend the concept of the measure of a matrix to encompass a measure induced by an arbitrary convex positive definite function. It is shown that this “modified” matrix measure has most of the properties of the usual matrix measure, and that many of the known applications of the usual matrix measure can therefore be carried over to the modified matrix measure. These applications include deriving conditions for a mapping to be a diffeomorphism on Rⁿ, and estimating the solution errors that result when a nonlinear network is approximated by a piecewise linear network. We also develop a connection between matrix measures and Liapunov functions. Specifically, we show that if V is a convex positive definite function and A is a Hurwitz matrix, then μ_V(A) < 0, if and only if V is a Liapunov function for the system $\text{x}\text{?}\text{= Ax}$. This linking up between matrix measures and Liapunov functions leads to some results on the existence of a “common” matrix measure μ_V(·) such that μ_V(A_i) < 0 for each of a given set of matrices A₁,…, A_m. Finally, we also give some results for matrices with nonnegative off-diagonal terms.     

Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q ₁,…,Q _r be quadratic forms with real coefficients. We prove that the set {(Q₁(x),?,Q_r(x)) | x ? \Bbb Z^s}\{(Q_1(x),\ldots ,Q_r(x))\,\vert\, x\in{\Bbb Z}^s\} is dense in \Bbb R^r{\Bbb R}^r , provided that the system Q ₁(x) = 0,…,Q _r (x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q ₁,…,Q _r are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form.     

Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods

The Kibble-Slepian formula expresses the exponential of a quadratic form Q(x) = x^tS(I + S)^?1x, S^t = S, in n variables x = col(x₁,…, x_n) as a series of products of Hermite polynomials, thus generalizing Mehler's formula. This extension is restricted, however, to the case where the diagonal elements of the symmetric matrix S are all unity. We derive the general formula for an arbitrary symmetric matrix S, where I + S is positive definite, using techniques familiar from the boson operator treatment of the harmonic oscillator in quantum mechanics.     

Existence of positive bounded solutions for some nonlinear elliptic systems

We prove some existence results of positive bounded continuous solutions to the semilinear elliptic system Δu=λp(x)g(v), Δv=μq(x)f(u) in domains D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f,g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the Kato class K(D).     

A new proof of a theorem of Harper on the Sperner-Erdös problem

Let P be a finite, partially ordered set and v a weight on P, i.e., a function v: P → R₊/{0. A subset F ? P) is called a k-family, if there are not c₀,…,c_k?F such that c₀ < … <c_k. Let d_k(P, v) = max {Σ_x?Fv(x); F is k-family. It is given a new proof of a theorem of Harper which states that d_k(P, v) = d_k(Q, w), if there is a flow morphism from (P, v) onto (Q, w).     

Lyapunov equations and Gram matrices

If p is a polynomial with all roots inside the unit disc and C its companion matrix, then the Lyapunov equation

$\text{X ? C}{}^1\text{XC = P}$

has a unique solution for every positive semidefinite matrix P. We characterize sets of vectors x₀,…,x_n?1 and y₀,…,y_n?1 such that X = G(x₀,…,x_n?1)= G(y₀,…, y_n?1)^-1. Geometrical connections between such bases and contractions with one- dimensional defect spaces are established.