Inclusion relations between certain sets of matrices

In this paper we study inclusion relations between the following four classes of matrices: normal matrices, matrices with equal spectral radius and spectral norm, matrices whose numerical range coincides with the convex polygon spanned by their eigenvalues, and matrices with equal numerical and spectral radii.     

Direct and inverse spectral problems for a class of non-self-adjoint periodic tridiagonal matrices

The spectral properties of a class of tridiagonal matrices are investigated. The reconstruction of matrices of this special class from given spectral data is also studied. Necessary and sufficient conditions for that reconstruction are found. The obtained results extend some results on the direct and inverse spectral problems for periodic Jacobi matrices and for some non-self-adjoint tridiagonal matrices.     

The numerical radius and specttural matrices

In this paper we investigate spectral matrices, i.e., matrices with equal spectral and numerical radii. Various characterizations and properties of these matrices are given.     

Concentration and convergence rates for spectral measures of random matrices

The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases.     

On asymptotic properties of matrix semigroups with an invariant cone

Recently, several research efforts showed that the analysis of joint spectral characteristics of sets of matrices can be simplified when these matrices share an invariant cone. We prove new results in this direction.We prove that the joint spectral subradius is continuous in the neighborhood of sets of matrices that leave an embedded pair of cones invariant.We show that both the averaged maximal spectral radius, as well as the maximal trace, where the maximum is taken on all the products of the same length t, converge towards the joint spectral radius when t increases, provided that the matrices share an invariant cone, and additionally one of them is primitive.     

On matrices having equal spectral radius and spectral norm

In this paper we characterize all nxn matrices whose spectral radius equals their spectral norm. We show that for n?3 the class of these matrices contains the normal matrices as a subclass.     

Generalizations of the spectral theorem for matrices. II. matrix polynomials over arbitrary fields

The spectral theorem for matrices is generalized to matrix polynomials over arbitrary fields. An associated theory of generalized interpolation with respect to a set of prime polynomials is developed, and Drazin inverses are used to identify the spectral components. These results are then used to state the spectral theorem for functions of real matrices.     

SP.n矩阵的分解

J.J．Johnson在1974年给出了P．n．P矩阵的谱性质具有一负特征值的充分条件及P．n矩阵的两个增性质．许多国内同仁也于近几年从分析偏负阵的Schur补入手，从特殊到一般，得到了一系列偏负阵的判定方法．本文在偏负阵判定方法的基础上，专门对偏负阵的分解进行论述，得出了对称偏负阵能进行三角分解和正交分解的充要条件．     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

To solving spectral problems for multiparameter polynomial matrices

An approach to solving spectral problems for multiparameter polynomial matrices based on passing to accompanying pencils of matrices is described. Also reduction of spectral problems for multiparameter pencils of complex matrices to the corresponding real problems is considered. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2006, pp. 212–231.     

Maximal eigenvalue and norm of a product of Toeplitz matrices. Study of a particular case

In this paper we describe the asymptotic behaviour of the spectral norm of the product of two finite Toeplitz matrices as the matrix dimension goes to infinity. These Toeplitz matrices are generated by functions with Fisher–Hartwig singularities of negative order. If these functions are positives the product of the two matrices has positive eigenvalues and it is known that the spectral norm is also the largest eigenvalue of this product.     

Spectra of infinite-dimensional sample covariance matrices

We study spectral functions of infinite-dimensional random Gram matrices of the form RR^T, where R is a rectangular matrix with an infinite number of rows and with the number of columns N → ∞, and the spectral functions of infinite sample covariance matrices calculated for samples of volume N → ∞ under conditions analogous to the Kolmogorov asymptotic conditions. We assume that the traces d of the expectations of these matrices increase with the number N such that the ratio d/N tends to a constant. We find the limiting nonlinear equations relating the spectral functions of random and nonrandom matrices and establish the asymptotic expression for the resolvent of random matrices. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 309–322, August, 2006.     

To solving problems of algebra for two-parameter matrices. VII

The paper discusses the method of rank factorization for solving spectral problems for two-parameter polynomial matrices. New forms of rank factorization, which are computed using unimodular matrices only, are suggested. Applications of these factorizations to solving spectral problems for two-parameter polynomial matrices of both general and special forms are presented. In particular, matrices free of the singular spectrum are considered. Conditions sufficient for a matrix to be free of the singular spectrum and also conditions sufficient for a basis matrix of the null-space to have neither the finite regular nor the finite singular spectrum are provided. Bibliography: 3 titles.     

Spectral analysis of large block random matrices with rectangular blocks

In this paper, we study the spectral properties of the large block random matrices when the blocks are general rectangular matrices. Under some moment assumptions of the underlying distributions, we prove the existence of the limiting spectral distribution (LSD) of the block random matrices. Further, we determine the Stieltjes transform of the LSD under the same moment conditions by demonstrating that it is the same as in the case where the underlying distributions are Gaussian.     

Optimizing matrix stability

Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matrices with nontrivial Jordan structure are rare in the space of all matrices, they appear naturally in spectral abscissa minimization.

     

多元线性混合模型方差分量矩阵的非负估计

本文研究了带有两个方差分量矩阵的多元线性混合模型方差分量矩阵的估计问题.对于平衡模型,给出了基于谱分解估计的一个方差分量矩阵的非负估计类.对于非平衡模型,给出了方差分量矩阵的广义谱分解估计类,讨论了与ANOVA估计等价的充要条件.同时,在广义谱分解估计的基础上给出了一种非负估计类,并讨论了其优良性.当具有较小二次风险的非负估计不存在时,从估计为非负的概率的角度考虑,将Kelly和Mathew(1993)提出的构造具有更小取负值概率的估计类的方法推广到本文的多元模型下,给出了较谱分解估计相比有更小取负值概率和更小风险的估计类.最后,模拟研究和实例分析表明文中理论结果有很好的表现.     

Optimal norms and the computation of joint spectral radius of matrices

The notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix. The finiteness conjecture (FC) claims that among the infinite products made from the elements of a given finite set of matrices, there is a certain periodic product, made from the repetition of the optimal product, whose rate of growth is maximal. FC has been disproved. In this paper it is conjectured that FC is almost always true, and an algorithm is presented to verify the optimality of a given product. The algorithm uses optimal norms, as a special subset of extremal norms. Several conjectures related to optimal norms and non-decomposable sets of matrices are presented. The algorithm has successfully calculated the spectral radius of several parametric families of pairs of matrices associated with compactly supported multi-resolution analyses and wavelets. The results of related numerical experiments are presented.     

Inverse problems for differential operators of any order on trees

Inverse spectral problems for ordinary differential operators of any order on compact trees are studied. As the main spectral characteristics, Weyl matrices, which generalize the Weyl m-function for the classical Sturm-Liouville operator are introduced and studied. A constructive solution procedure for the inverse problem based on Weyl matrices is suggested, and the uniqueness of the solution is proved. The reconstruction of differential equations from discrete spectral characteristics is also considered.     

On irreducible factorizations of rational matrices and their applications

This paper is an extension of our studies of the computational aspects of spectral problems for rational matrices pursued in previous papers. Methods of solution of spectral problems for both one-parameter and two-parameter matrices are considered. Ways of constructing irreducible factorizations (including minimal factorizations with respect to the degree and size of multipliers) are suggested. These methods allow us to reduce the spectral problems for rational matrices to the same problems for polynomial matrices. A relation is established between the irreducible factorization of a one-parameter rational matrix and its irreducible realization used in system theory. These results are extended to the case of two-parameter rational matrices. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 117–156. This work was carried out during our visit to Sweden under the financial support of the Chalmer University of Technology in Góterborg and the Institute of Information Processing of the University of Umeă. Translated by V. N. Kublanovskaya.     

由特征值唯一确定的实3×3Hankel矩阵(英文)

本文研究了由特征值唯一确定的3×3实Hankel矩阵.借助于M.Fielder[1]的结论并经过细致的讨论,得到3×3实Hankel矩阵由其特征值唯一确定的充分必要条件,刻画了3×3实Hankel矩阵的一种特征值性质.