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 Semigroup Generated by the Similarity Class of a Singular Matrix

Let A be a singular matrix of M _n (𝕂), where 𝕂 is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of ( _n (𝕂), ×) generated by the similarity class of A is the set of matrices of M _n (𝕂) with a rank lesser than or equal to that of A.     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

More refined enumerations of alternating sign matrices

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general “d-refined” enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_n,i,j that enumerate such matrices. We give a conjectural explicit formula for A_n,i,j and formulate several other conjectures about the sufficiency of the linear equations to determine the A_n,i,j's and about an extension of the linear equations to the general d-refined enumerations.     

An implicit QR algorithm for symmetric semiseparable matrices

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n³) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

The Combinatorics of a Class of Representation Functions

In the study of the irreducible representations of the unitary groupU(n), one encounters a class of polynomials defined onn²indeterminatesz_ij, 1i, jn, which may be arranged into ann×nmatrix arrayZ=(z_ij). These polynomials are indexed by double Gelfand patterns, or equivalently, by pairs of column strict Young tableaux of the same shape. Using the double labeling property, one may define a square matrixD(Z), whose elements are the double-indexed polynomials. These matrices possess the remarkable “group multiplication property”D(XY)=D(X) D(Y) for arbitrary matricesXandY, even though these matrices may be singular. ForZ=UU(n), these matrices give irreducible unitary representations ofU(n). These results are known, but not always fully proved from the extensive physics literature on representation of the unitary groups, where they are often formulated in terms of the boson calculus, and the multiplication property is unrecognized. The generality of the multiplication property is the key to understanding group representation theory from the purview of combinatorics. The combinatorial structure of the general polynomials is expected to be intricate, and in this paper, we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums. These special polynomials are denoted byL_{α, β}(Z), whereαandβare integral vectors representing the row sums and column sums of a class of integral matrices. We present a combinatorial interpretation of the multiplicative properties of these polynomials. We also point out the connections with MacMahon's Master Theorem and Schwinger's inner product formula, which is essentially equivalent to MacMahon's Master Theorem. Finally, we give a formula for the double Pfaffian, which is crucial in the studies of the generating function of the 3n−jcoefficients in angular momentum theory. We also review the background of the general polynomials and give some of their properties.     

On the convergence of averaging Hermite interpolators

We investigate two sequences of polynomial operators, H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H_{2n − 1}(f, x). If H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x) are based on the zeros of the jacobi polynomials P_n^(α,β)(x), their convergence behaviour is similar to that of H_{2n − 1}(f;, x). If they are based on the zeros of (1 − x²)T_{n − 2}(x), their convergence behaviour is better, in some sense, than that of H_{2n − 1}(f, x).     

Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z ₀ be a subset of Z such than n∈Z ₀ implies n + 1 ∈Z ₀. Denote the space of all compactly supported distributions by D′, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G _n and H _n , n∈Z ₀, in D′, define the corresponding nonstationary nonhomogeneous refinement equation Φ _n =H _n *Φ _n+1 (A·)+G _n for all n∈Z ₀ where Φ _n , n∈Z ₀, is in a bounded set of D′. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ _n , n∈Z ₀, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations for all n∈Z ₀ where the matrices S _n and the vectors , n∈Z ₀, can be constructed explicitly from H _n and G _n respectively. The results above are still new even for stationary nonhomogeneous refinement equations. Received December 30, 1999, Accepted June 15, 2000     

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S _n (𝔽) denote the vector space of all n?×?n symmetric matrices over 𝔽. We characterize i. subspaces of S _n (𝔽) all whose elements have rank at most two where n???3,

ii. linear maps from S _m (𝔽) to S _n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n???3 and |𝔽|?≠?2.

     

Eigenvalues of products of matrices

Let A and B be n?×?n matrices over an algebraically closed field F. The pair ( A,?B ) is said to be spectrally complete if, for every sequence c₁,…,c_n ∈F such that det (AB)=c₁ ,…,c_n , there exist matrices A′,B,′∈F,^n×n similar to A,?B, respectively, such that A′B′ has eigenvalues c₁,…,c_n . In this article, we describe the spectrally complete pairs. Assuming that A and B are nonsingular, the possible eigenvalues of A′B′ when A′ and B′ run over the sets of the matrices similar to A and B, respectively, were described in a previous article.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Perturbation Analysis for the Eigenvalue Problem of a Formal Product of Matrices

We study the perturbation theory for the eigenvalue problem of a formal matrix product A ₁ ^{s 1} ··· A _p ^{s p}, where all A _k are square and s _k {–1, 1}. We generalize the classical perturbation results for matrices and matrix pencils to perturbation results for generalized deflating subspaces and eigenvalues of such formal matrix products. As an application we then extend the structured perturbation theory for the eigenvalue problem of Hamiltonian matrices to Hamiltonian/skew-Hamiltonian pencils.     

Maps preserving square-zero matrices on the algebra consisting of all upper triangular matrices

Let F be a field, T _n (F) (respectively, N _n (F)) the matrix algebra consisting of all n × n upper triangular matrices (respectively, strictly upper triangular matrices) over F. A ∈ T _n (F) is said to be square zero if A ² = 0. In this article, we firstly characterize non-singular linear maps on N _n (F) preserving square-zero matrices in both directions, then by using it we determine non-singular linear maps on T _n (F) preserving square-zero matrices in both directions.     

A new investigation of the extended Krylov subspace method for matrix function evaluations

For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd.     

Structure Theory for Grade Three Perfect Ideals Associated with Some Matrices

Kang, Cho, and Ko gave a structure theorem for some classes of perfect ideals of grade 3 which are linked to almost complete intersections by a regular sequence. We define a 5 × 9 matrix f determined by three matrices A, T, and Y and construct a class of perfect ideals 𝒦₄(f) of grade 3 defined by f. This contains a class of perfect ideals of grade 3 minimally generated by five elements which are not Gorenstein. We give a structure theorem for some classes of perfect ideals of grade 3 linked to 𝒦₄(f) by a regular sequence in 𝒦₄(f).     

Jensen matrix inequalities and direct sums

Let A, X and Y be n-by-n complex matrices such that A is positive semi-definite and X, Y are contractions. We prove that if f is an increasing convex function on [0, ∞) such that f(0) ≤ 0, then the eigenvalues of f(|X*AY|) are dominated by those of X*f(A)X⊕Y* f(A)Y. Several related results are considered.     

Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH: ∫^π_−π K{f(λ)} dλ=cagainstA: ∫^π_−π K{f(λ)} dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testT_nbased on ∫^π_−π K{f(λ)} dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), and we define an efficacy ofT_nunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf₄^Zofz(t). If it does not depend onf₄^Z, we say thatT_nis non-Gaussian robust. We will give sufficient conditions forT_nto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results.     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

On the Properties of Accretive-Dissipative Matrices

Let A be a complex n × n matrix, and let A = B + iC, B = B*, C = C* be its Toeplitz decomposition. Then A is said to be (strictly) accretive if B > 0 and (strictly) dissipative if C > 0. We study the properties of matrices that satisfy both these conditions, in other words, of accretive-dissipative matrices. In many respects, these matrices behave as numbers in the first quadrant of the complex plane. Some other properties are natural extensions of the corresponding properties of Hermitian positive-definite matrices.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 832–843.Original Russian Text Copyright ©2005 by A. George, Kh. D. Ikramov.