Invariant functionals for random matrices

A new approach to the study of the Lyapunov exponents of random matrices is presented. It is proved that, under general assumptions, any family of nonnegative matrices possesses a continuous concave positively homogeneous invariant functional (“antinorm”) on ℝ₊^d. Moreover, the coefficient corresponding to an invariant antinorm equals the largest Lyapunov exponent. All conditions imposed on the matrices are shown to be essential. As a corollary, a sharp estimate for the asymptotics of the mathematical expectation for logarithms of norms of matrix products and of their spectral radii is derived. New upper and lower bounds for Lyapunov exponents are obtained. This leads to an algorithm for computing Lyapunov exponents. The proofs of the main results are outlined.     

Some closer bounds of Perron root basing on generalized Perron complement

This paper is concerned with the bounds of the Perron root ρ(A) of a nonnegative irreducible matrix A. Two new methods utilizing the relationship between the Perron root of a nonnegative irreducible matrix and its generalized Perron complements are presented. The former method is efficient because it gives the bounds for ρ(A) only by calculating the row sums of the generalized Perron complement P_t(A/A[α]) or even the row sums of submatrices A[α],A[β],A[α,β] and A[β,α]. And the latter gives the closest bounds (just in this paper) of ρ(A). The results obtained by these methods largely improve the classical bounds. Numerical examples are given to illustrate the procedure and compare it with others, which shows that these methods are effective.     

Asymptotic formula for the decay of correlations in the stochastic model of planar rotators at high temperature

We consider the stochastic model of planar rotators x(t)={x_k(t), k∈Z^d}, t≥0, x_k(t)∈T¹, at high temperature. For the decay of correlations <f_A(x(0)), g_A+k(t) (x(t))>, the asymptotic formula is obtained at t→∞, k(t)→∞, k(t)∈Z^d. The basic methods we used are the spectral analysis of the Markov semigroup generator and the saddle-point method. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 67–80.     

Bounds for norms of the matrix inverse and the smallest singular value

In the first part, we obtain two easily calculable lower bounds for ‖A^-1‖, where ‖·‖ is an arbitrary matrix norm, in the case when A is an M-matrix, using first row sums and then column sums. Using those results, we obtain the characterization of M-matrices whose inverses are stochastic matrices. With different approach, we give another easily calculable lower bounds for ‖A^-1‖_∞ and ‖A^-1‖₁ in the case when A is an M-matrix. In the second part, using the results from the first part, we obtain our main result, an easily calculable upper bound for ‖A^-1‖₁ in the case when A is an SDD matrix, thus improving the known bound. All mentioned norm bounds can be used for bounding the smallest singular value of a matrix.     

Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K = D A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius p(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.     

The total reducibility order of a polynomial in two variables

LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ _i=1 ^n(λ) A _iλ ^k _μ whereA _iλ are distinct primes. Let ϱ_λ(A) =n(λ) − 1 and let ρ(A) = Σ_λɛσ(A)ϱ_λ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.     

Schur multiplier characterization of a class of infinite matrices

Let B _w (ℓ ^p ) denote the space of infinite matrices A for which A(x) ∈ ℓ ^p for all x = {x _k } _k=1^∞ ∈ ℓ ^p with |x _k | ↘ 0. We characterize the upper triangular positive matrices from B _w (ℓ ^p ), 1 < p < ∞, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.     

Nonnegative matrices having nonnegative Drazin pseudoinverses

Necessary and sufficient conditions for nonnegative matrices having nonnegative Drazin pseudoinverses are obtained. A decomposition theorem which characterizes the class of all nonnegative matrices with nonnegative Drazin pseudoinverses is proved, thus answering a question raised by several people. It is also shown that if a row (or column) stochastic matrix has a nonnegative Drazin pseudoinverse A(^d), then A(^d) is some power of A. These results extend known results for nonnegative group-monotone matrices.     

Doubly stochastic matrix equations

It is shown that for real,m x n matricesA andB the system of matrix equationsAX=B, BY=A is solvable forX andY doubly stochastic if and only ifA=BP for some permutation matrixP. This result is then used to derive other equations and to characterize the Green’s relations on the semigroup Ω _n of alln x n doubly stochastic matrices. The regular matrices in Ω _n are characterized in several ways by use of the Moore-Penrose generalized inverse. It is shown that a regular matrix in Ω _n is orthostochastic and that it is unitarily similar to a diagnonal matrix if and only if it belongs to a subgroup of Ω _n . The paper is concluded with extensions of some of these results to the convex setS _n of alln x n nonnegative matrices having row and column sums at most one. His research was supported by the N. S. F. Grant GP-15943.     

Some new bounds on the spectral radius of matrices

A new lower bound on the smallest eigenvalue τ(AB) for the Fan product of two nonsingular M-matrices A and B is given. Meanwhile, we also obtain a new upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B. These bounds improve some results of Huang (2008) [R. Huang, Some inequalities for the Hadamard product and the Fan product of matrices, Linear Algebra Appl. 428 (2008) 1551-1559].     

Extensions of the Hadamard determinant theorem

Denote byH _n the set ofn byn, positive definite hermitian matrices. Hadamard proved thath(A)≧det(A) for allA∈H _n, whereh(A) is the product of the main diagonal elements ofA. Subsequently, M. Marcus showed that per(A)≧h(A) for allA∈H _n. This article contains a result for all generalized matrix functions from which it follows thath(A)≧(per(A^1/n )) ⁿ ,A∈H _n.     

Some simple Haar-type wavelets in higher dimensions

An orthonormal wavelet system in ℝ^d, d ∈ ℕ, is a countable collection of functions {ψ _j,k ^ℓ }, j ∈ ℤ, k ∈ ℤ^d, ℓ = 1,..., L, of the form that is an orthonormal basis for L² (ℝ^d), where a ∈ GL_d (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L = d = 1, ψ¹(x) = ψ(x) = κ[0,1/2)^(x) − κ_[l/2,1) ^(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate products Φ(x₁, x₂, ..., x_d) = φ₁ (x₁)φ₂(x₂) ... φ_d(x_d) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems. For example, if a = ( _1-1 ^{1 1} ) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed in [7] and involve dilations by matrices that are products of the form a^jb, j ∈ ℤ, where a ∈ GL_d(ℝ) has some “expanding” property and b belongs to a group of matrices in GL_d(ℝ) having |det b| = 1.     

Sharp bounds on the spectral radius of a nonnegative matrix

We give upper and lower bounds for the spectral radius of a nonnegative matrix using its row sums and characterize the equality cases if the matrix is irreducible. Then we apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix. Some known results in the literature are generalized and improved.     

NON-EMPTY CROSS-2-INTERSECTING FAMILIES OF SUBSETS

Let Cdenote the set of all k-subests of an n-set.Assume Alohtain in Ca,and A lohtain in (A,B) is called a cross-2-intersecting family if |A B≥2 for and A∈A,B∈B.In this paper,the best upper bounds of the cardinalities for non-empty cross-2-intersecting familles of a-and b-subsets are obtained for some a and b,A new proof for a Frankl-Tokushige theorem[6] is also given.     

Zero-Eigenvalue Eigenfunctions for Differences of Elliptic Relativistic Calogero-Moser Hamiltonians

Letting A_l(x) denote the commuting analytic difference operators of elliptic relativistic Calogero-Moser type, we present and study zero-eigenvalue eigenfunctions for the operators A_l(x) − A_l(−y) (with l = 1, 2,..., N and x, y ∈ ℂ ^N). The eigenfunctions are products of elliptic gamma functions. They are invariant under permutations of x₁,..., x_N and y₁,..., y_N and under interchange of the step-size parameters. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 31–41, January, 2006.     

On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A₁°A₂°?°A_k)?ρ(A₁A₂?A_k) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A°B‖?ρ(A^TB) for nonnegative matrices, which improves Schur’s classical inequality ‖A°B‖?‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.     

Residual behavior of induced maps

Consider (X,F, μ,T) a Lebesgue probability space and measure preserving invertible map. We call this a dynamical system. For a subsetA ∈F. byT _A:A →A we mean the induced map,T _A(x)=T^rA(x)(x) wherer _A(x)=min{i〉0:T ⁱ(x) ∈A}. Such induced maps can be topologized by the natural metricD(A, A’) = μ(AΔA’) onF mod sets of measure zero. We discuss here ergodic properties ofT _A which are residual in this metric. The first theorem is due to Conze.Theorem 1 (Conze):For T ergodic, T _A is weakly mixing for a residual set of A.Theorem 2:For T ergodic, 0-entropy and loosely Bernoulli, T _A is rank-1, and rigid for a residual set of A.Theorem 3:For T ergodic, positive entropy and loosely Bernoulli, T _A is Bernoulli for a residual set of A.Theorem 4:For T ergodic of positive entropy, T _A is a K-automorphism for a residual set of A. A strengthening of Theorem 1 asserts thatA can be chosen to lie inside a given factor algebra ofT. We also discuss even Kakutani equivalence analogues of Theorems 1–4.     

Multiparametric estimating equations

Summary Let {p(x, θ): θ∈Θ} be a family of densities where θ=(θ₁,θ₂), being θ₁ ∈ Θ₁ ak-dimensional parameter of interest, θ₂ ∈ Θ₂ a nuisance parameter and Θ=Θ₁×Θ₂. To estimate θ₁, vector estimating equations g(x,θ₁)=(g₁(x,θ₁),...,g_k(x,θ₁))=0 are considered. The standardized form of g(x,θ₁) is defined as g_s=(E_θ(∂g/∂θ′₁))⁻¹g. Then, within the classG ₁ of unbiased equations (i.e. satisfying E_θ(g)=0 (θ∈Θ)), an equationg ^*=0 is said to be optimum if the covariance matrices ofg _s andg _s ^* are such that is non-negative definite for allg∈ G ₁ and θ∈Θ. Sufficient conditions for optimality are discussed and, in particular, conditions for the optimality of the maximum conditional likelihood equation are analyzed. Special attention is given to non-regular cases. In addition, measures of the information about θ₁ contained in an estimating equation are presented and a Rao-Blackwell theorem is given. CIENES     

Resolvent Estimates for Operators Belonging to Exponential Classes

For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that s _n (A) = O(exp(-an^α)), where s _n (A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)⁻¹ of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).