Characterizations of products of symmetric matrices

Characterizations are obtained for matrices C of the form C = AΣ, where A, Σ are n×n matrices over the real field such that A is symmetric and C is nonnegative definite. Among others, a proof of recent generalization of Cochran's theorem is given.     

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.     

A characterization of Lévy probability distribution functions on Euclidean spaces

In 1937, Paul Lévy proved two theorems that characterize one-dimensional distribution functions of class L. In 1972, Urbanik generalized Lévy's first theorem. In this note, we generalize Lévy's second theorem and obtain a new characterization of Lévy probability distribution functions on Euclidean spaces. This result is used to obtain a new characterization of operator stable distribution functions on Euclidean spaces and to show that symmetric Lévy distribution functions on Euclidean spaces need not be symmetric unimodal.     

Produkte positiver elemente in formal-reellen Jordan-Algebren

A generalization of the Rayleigh quotient defined for real symmetric matrices to the elements of a formally real Jordan algebra is used here to give a generalization to formally real Jordan algebras of the theorem that for any real symmetric matrix C with tr C > 0 there are positive definite real symmetric matrices A and B with C = AB + BA.     

Conjugate cone characterization of positive definite and semidefinite matrices

Positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in Rⁿ. These results are obtained by generalizing Moreau's polar decomposition to a conjugate decomposition. Some typical results are: The matrix A is positive definite if and only if for some closed convex cone K, A is positive definite on K and (A+A^T)^?1 exists and is semidefinite on the polar cone K°. The matrix A is positive semidefinite if and only if for some closed convex cone K such that either K is polyhedral or (A+A^T)(K) is closed, A is positive semidefinite on both K and the conjugate cone K^A={s∣x^T(A+ A^T)s?0?x∈K}, and (A+A^T)x=0 for all x in K such that x^TAx=0.     

On strong solutions for positive definite jump diffusions

We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics.Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the αth positive semidefinite power of the process itself with 0.5<α<1 and obtain existence conditions for them. In the case of a diffusion coefficient which is linear in the process we likewise get a positive definite analogue of the univariate GARCH diffusions.     

Projection theorems for hitting probabilities and a theorem of Littlewood

Littlewood (Proc. London Math. Soc. (2), 28 1928, 383–394) showed that a positive superharmonic function u on the unit disc has radial limits a.e. Using techniques due to Doob this result is extended to all rank one symmetric spaces. In addition simplifications are obtained of Doob's (Ann. Inst. Fourier (Grenoble), 15 1965, 113–135) proof of normal convergence a.e. of a positive superharmonic function on a half space. The symmetric space analogue of this half space result is also obtained. The methods used are shown to fail for the potential theory on $\text{R}$ⁿ associated with Δu = αu (α > 4 0). It is an open question as to whether Littlewood's theorem holds in this context.     

Generalized qd algorithm for block band matrices

The generalized qd algorithm for block band matrices is an extension of the block qd algorithm applied to a block tridiagonal matrix. This algorithm is applied to a positive definite symmetric block band matrix. The result concerning the behavior of the eigenvalues of the first and the last diagonal block of the matrix containing the entries q ^(k) which was obtained in the tridiagonal case is still valid for positive definite symmetric block band matrices. The eigenvalues of the first block constitute strictly increasing sequences and those of the last block constitute strictly decreasing sequences. The theorem of convergence, given in Draux and Sadik (Appl Numer Math 60:1300?C1308, 2010), also remains valid in this more general case.     

Proximal point algorithm with Schur decomposition on the cone of symmetric semidefinite positive matrices

In this work, we propose a proximal algorithm for unconstrained optimization on the cone of symmetric semidefinite positive matrices. It appears to be the first in the proximal class on the set of methods that convert a Symmetric Definite Positive Optimization in Nonlinear Optimization. It replaces the main iteration of the conceptual proximal point algorithm by a sequence of nonlinear programming problems on the cone of diagonal definite positive matrices that has the structure of the positive orthant of the Euclidian vector space. We are motivated by results of the classical proximal algorithm extended to Riemannian manifolds with nonpositive sectional curvature. An important example of such a manifold is the space of symmetric definite positive matrices, where the metrics is given by the Hessian of the standard barrier function −lndet(X). Observing the obvious fact that proximal algorithms do not depend on the geodesics, we apply those ideas to develop a proximal point algorithm for convex functions in this Riemannian metric.     

A characterization of 3-connected graphs containing a given graph

It is shown, as a complement to Tutte's theorem, that for a given 3-connected graph K which is not a wheel, a graph G is 3-connected and has a subgraph contractible to K if and only if G can be obtained from K by a finite sequence of line-additions and 3-point-splittings.     

Cohomologie des groupes localement compacts et produits tensoriels continus de représentations

The following is an expository paper, containing few and sometimes incomplete proofs, on continuous tensor products of Hilbert spaces and of group representations, and on the irreducibility of the latter; the principal results in the last direction are due to Verchik, Gelfand, and Graiev. The theory of continuous tensor products of Hilbert spaces, based on a fundamental theorem of Araki and Woods, is closely related to that of conditionally positive definite functions; it relies on the technique of symmetric Hilbert spaces, which also can be used to give a new proof of the classical Lévy-Khinchin formula (see A. Guichardet, (1973). J. Multiv.3 249–261.). Another basic tool for what follows is the 1-cohomology of unitary representations of locally compact groups; here, the main results are due to P. Delorme; let us mention, for instance, his results for the case of a group G containing a compact subgroup K such that $\text{L}{}^1\text{(Kβ}\text{G}\text{K}\text{)}$ is commutative, using a Lévy-Khinchin's type formula for K-invariant functions due to Gangolli, Faraut, and Harzallah. We add that the results exposed in that paper should have interesting connections with the central limit theorems à la Parthasarathy-Schmidt (see K. Parthasarathy, (1974). J. Multiv. Anal.4 123–149).     

Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups

The theory of symmetric local semigroups due to A. Klein and L. Landau (J. Funct. Anal.44 (1981), 121–136) is generalized to semigroups indexed by subsets of $\text{R}$ⁿ for n > 1. The result implies a similar result of A. E. Nussbaum (J. Funct. Anal.48 (1982), 213–223). It is further generalized to semigroups that are symmetric local in some directions and unitary in others. The results are used to give a simple proof of A. Devinatz's (Duke Math. J.22 (1955), 185–192) and N. I. Akhiezer's (“the Classical Moment Problem and Some Related Questions,” Hafner, New York, 1965) generalization of a theorem of Widder concerning the representation of functions as Laplace integrals. This result is extended to the representation as a Laplace integral of a function taking values in $\text{B}$($\text{R}$), the set of bounded linear operators on a Hilbert space $\text{R}$. Also, a theorem is proved encompassing both the result of Devinatz and Akhiezer, and Bochner's theorem on the representation of positive definite functions as Fourier integrals.     

A combinatorial proof of Marstrand’s theorem for products of regular Cantor sets

In a paper from 1954 Marstrand proved that if K⊂R² has a Hausdorff dimension greater than 1, then its one-dimensional projection has a positive Lebesgue measure for almost all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^1+α, α>0, for which the sum of their Hausdorff dimension is greater than 1.     

Group-invariant analogues of Hadamard's inequality

A wide class of inequalities for the determinant and other real-valued functions of an n × n complex Hermitian (or real symmetric) matrix H≡(h_jk) may be obtained by generalizing Marshall and Olkin's proof of Hadamard's inequality

$\text{det}\text{H?}\text{∏}\text{j=1}\text{n}\text{h}{}_{\mathrm{jj}}$

for positive definite (pd) H. We shall see that each subgroup G of the group U_n of n x n unitary matrices not only determines an analogue of (1) for det H, but also provides inequalities for a large family of unitarily invariant functions of H (not necessarily pd).     

Bernstein theorems and transformations of correlation measures in statistical physics

We study the class of endomorphisms of the cone of correlation functions generated by probability measures. We consider algebraic properties of the products (·, ?) and the maps K, K ^?1 which establish relationships between the properties of functions on the configuration space and the properties of the corresponding operators (matrices with Boolean indices): F(γ) → F?(γ) = {F(α?β)}α,β?γ. For the operators F?(γ) and F?(γ), we prove conditions which ensure that these operators are positive definite; the conditions are given in terms of complete or absolute monotonicity properties of the function F(γ).     

Complementarity properties of the Lyapunov transformation over symmetric cones

The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX +XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V . Given any element a ∈ V , we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0 -property for La and show that La has the R0 -property if and only if a is invertible. Finally, we provide La with some characterizations of the E0 -property and the nondegeneracy property.     

Non-skew symmetric orthogonal matrices with constant diagonals

A matrix C of order n is orthogonal if CC^T=dI. In this paper, we restrict the study to orthogonal matrices with a constant m > 1 on the diagonal and ±1's off the diagonal. It is observed that all skew symmetric orthogonal matrices of this type are constructed from skew symmetric Hadamard matrices and vice versa. Some simple necessary conditions for the existence of non-skew orthogonal matrices are derived. Two basic construction techniques for non-skew orthogonal matrices are given. Several families of non-skew orthogonal matrices are constructed by applying the basic techniques to well-known combinatorial objects like balanced incomplete block designs. It is also shown that if m is even and n=0 (mod 4), then an orthogonal matrix must be skew symmetric. The structure of a non-skew orthogonal matrix in the special case of m odd,n=2 (mod 4) and m?1/6n is also studied in detail. Finally, a list of cases with n?50 is given where the existence of non-skew orthogonal matrices are unknown.     

Upper eigenvalue bounds for pencils of matrices

Eigenvalue bounds are obtained for pencils of matrices A ? vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods, where B represents the approximate factorization of A. The upper bounds obtained depend on the “connectivity” structure of the matrices involved, which enters through matrix graph considerations; in addition, a more classical argument is used to obtain a lower bound. Potential applications of these results include a partial confirmation of Gustafsson's conjecture concerning the nonnecessity of Axelsson's perturbations.     

Linear maps that preserve singular and nonsingular matrices

Let M (n,K) be the algebra of n × n matrices over an algebraically closed field K and T:M (n,K)→M (n,K) a linear transformation with the property that T maps nonsingular (singular) matrices to nonsingular (singular) matrices. Using some elementary facts from commutative algebra we show that T is nonsingular and maps singular matrices to singular matrices (T is nonsingular or T maps all matrices to singular matrices). Using these results we obtain Marcus and Moyl's characterization [T(x) = UXVorU^tXV for fixed U and V] from a result of Dieudonné's. Examples are given to show the hypothesis of algebraic closure in necessary.     

A classification of 4-connected graphs

Having observed Tutte's classification of 3-connected graphs as those attainable from wheels by line addition and point splitting and Hedetniemi's classification of 2-connected graphs as those obtainable from K₂ by line addition, subdivision and point addition, one hopes to find operations which classify n-connected graphs as those obtainable from, for example, K_n+1. In this paper I give several generalizations of the above operations and use Halin's theorem to obtain two variations of Tutte's theorem as well as a classification of 4-connected graphs.