Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities

We construct six unitary trace invariants for 2×2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We have discovered a curious trace identity for two unit-speed one-parameter subgroups of Sp(1). A modification gives an infinite family of trace identities for quaternions as well as for 2×2 complex matrices. We were not able to locate these identities in the literature. We prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field. Finally we consider the problem of describing the semi-algebraic set of pairs (X,Y) of quaternionic n×n matrices which are simultaneously triangularizable. Even the case n=2, which we analyze in more detail, remains unsolved.     

Hybrid functions approach for optimal control of systems described by integro-differential equations

In this paper, a new numerical method for solving the optimal control of a class of systems described by integro-differential equations with quadratic performance index is presented. This optimization problem plays an important role in describing the dynamics of an elastic aircraft with allowance for non-steady flow past its profile. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the solution of optimization problem to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source

We consider ensembles of random Hermitian matrices with a distribution measure determined by a polynomial potential perturbed by an external source. We find the genus-zero algebraic function describing the limit mean density of eigenvalues in the case of an anharmonic potential and a diagonal external source with two symmetric eigenvalues. We discuss critical regimes where the density support changes the connectivity or increases the genus of the algebraic function and consequently obtain local universal asymptotic representations for the density at interior and boundary points of its support (in the generic cases). The investigation technique is based on an analysis of the asymptotic properties of multiple orthogonal polynomials, equilibrium problems for vector potentials with interaction matrices and external fields, and the matrix Riemann-Hilbert boundary value problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 34–57, April, 2009.     

MP matrices

MP matrices are those real matrices which possess a nonnegative, nonsingular l-inverse. The difficult task of describing the class of MP matrices is investigated.     

On conditions for permutability of Toeplitz and Hankel matrices

The problem of describing pairs of commuting matrices (T, H), where T and H are a Toeplitz and a Hankel matrix, respectively, is examined. Several families of such pairs are indicated.     

Algebraic Methods for Condiagonalization Under Consimilarity of Quaternion Matrices in Quaternionic Quantum Mechanics

By means of complex representation and real representation of quaternion matrices, paper [7] studied the problem of diagonalization of quaternion matrices. This paper introduces two new complex representation and real representation of quaternion matrices, studies the problem of condiagonalization under consimilarity of quaternion matrices, and gives two algebraic methods for the condiagonalization under consimilarity of quaternion matrices.     

Doubly coprime representation of linear systems and its application to simultaneous stabilization

This paper develops a new representation theory of multivariabletime-invariant linear systems based on the well-known coprimefactorizations of their transfer matrices. This representationfully uses the algebraic structure of coprime factorizations,and hence has a number of advantages for describing variousproperties of linear systems in simpler and/or more compactforms than the usual transfer matrix representation. In theframework of this new representation theory, various stabilityand stabilizability properties of linear systems are characterizedand finally a simultaneous stabilization problem for a givenset of linear systems is examined.     

On the Dimension of the Set of Two-View Multi-Homography Matrices

It is shown that the set of all multi-homography matrices describing I-element families of interdependent homographies between two views has dimension 4I + 7.     

On extremum properties of orthogonal quotients matrices

In this paper we explore the extremum properties of orthogonal quotients matrices. The orthogonal quotients equality that we prove expresses the Frobenius norm of a difference between two matrices as a difference between the norms of two matrices. This turns the Eckart-Young minimum norm problem into an equivalent maximum norm problem. The symmetric version of this equality involves traces of matrices, and adds new insight into Ky Fan’s extremum problems. A comparison of the two cases reveals a remarkable similarity between the Eckart-Young theorem and Ky Fan’s maximum principle. Returning to orthogonal quotients matrices we derive “rectangular” extensions of Ky Fan’s extremum principles, which consider maximizing (or minimizing) sums of powers of singular values.     

On two particular cases of solving the normal Hankel problem

The normal Hankel problem is one of characterizing all the complex matrices that are normal and Hankel at the same time. The matrix classes that can contain normal Hankel matrices admit a parameterization by real 2 × 2 matrices with determinant one. Here, the normal Hankel problem is solved in the case where the characteristic matrix of a given class is an order two Jordan block for the eigenvalue 1 or ?1.     

Fast iteration method in the problem of waves interacting with a set of thin screens

A new iteration method is proposed for the wave equation describing the scattering of a harmonic wave from an arbitrary configuration in the form of an array of thin straight barriers. The problem is reduced to a system of boundary integral equations, which are discretized by applying the Belotserkovskii-Lifanov method. In discrete form, a finite number of systems with Toeplitz matrices (the number of systems is equal to the number of barriers) are solved at each iteration step by applying special fast methods. The algorithm is tested on several geometries, and its convergence in these cases is analyzed.     

Two new operators on fuzzy matrices

The fuzzy matrices are successfully used when fuzzy uncertainty occurs in a problem. Fuzzy matrices become popular for last two decades. In this paper, two new binary fuzzy operators ⊕ and ⊙ are introduced for fuzzy matrices. Several properties on ⊕ and ⊙ are presented here. Also, some results on existing operators along with these new operators are presented.     

On the properties of a new tensor product of matrices

Previously, the author introduced a new tensor product of matrices according to which the matrix of the discrete Walsh-Paley transform can be represented as a power of the second-order discrete Walsh transform matrix H with respect to this product. This power is an analogue of the representation of the Sylvester-Hadamard matrix in the form of a Kronecker power of H. The properties of the new tensor product of matrices are examined and compared with those of the Kronecker product. An algebraic structure with the matrix H used as a generator element and with these two tensor products of matrices is constructed and analyzed. It is shown that the new tensor product operation proposed can be treated as a convenient mathematical language for describing the foundations of discrete Fourier analysis.     

Linear programming based decomposition approach in evaluating priorities from pairwise comparisons and error analysis

One of the most difficult issues in many real-life decisionmaking problems is how to estimate the pertinent data. An approach which uses pairwise comparisons was proposed by Saaty and is widely accepted as an effective way of determining these data. Suppose that two matrices with pairwise comparisons are available. Furthermore, suppose that there is an overlapping of the elements compared in these two matrices. The problem examined in this paper is how to combine the comparisons of the two matrices in order to derive the priorities of the elements considered in both matrices. A simple approach and a linear programming approach are formulated and analyzed in solving this problem. Computational results suggest that the LP approach, under certain conditions, is an effective way for dealing with this problem. The proposed approach is of critical importance because it can also result in a reduction of the total required number of comparisons.The author would like to thank Professors Stuart H. Mann, Pennsylvania State University, and Panos M. Pardalos, University of Florida, for their support and valuable comments during the early stages of this research.     

Inversion and canonization of block matrices

The paper is concerned with the problem of inverting block matrices to which the well-known Frobenius— Schur formulas are not applicable. These can be square matrices with four noninvertible square or rectangular blocks as well as square or rectangular matrices with two blocks. With regard to rectangular matrices, the results obtained are a further step in the development of the canonization method, which is used for solving arbitrary matrix equations.     

干扰具有模型的抗干性问题

自从七十年代初期,Wonham教授提出干扰解耦问题以来,许多人讨论了具有任意干扰的线性系统用状态反馈实现抗干扰的问题,给出了实现抗干扰的充要条件。其中大量的工作,是讨论和具体构造Wonbam教授提出的(A,B)最大不变子空间。但有     

On the Matrix Method in the Theory of Wave Propagation in Layered Biot Media

Wave propagation in porous Biot media with homogeneous isotropic layers is investigated by the matrix method. In order to use this method, matrices describing porous layers and half-spaces are established. On the basis of these matrices, the wave fields in layered porous media are derived and examined. In this paper, matrices of inhomogeneous porous layers are also determined, and these matrices are represented by convergent series. Bibliography: 9 titles.     

Properties of asymptotic coefficients of the functions of the continuous spectrum of a lightguide

The asymptotic coefficients of the eigenfunctions of the continuous spectrum of a periodic lightguide are investigated. Integral equations, describing the spectrum of scattering matrices and other symplectic matrices that connect the asymptotic coefficients, are obtained. The directed motion of the eigenvalues of the scattering matrices and of certain groups of eigenvalues of symplectic matrices, under a parametric monotonic change in the medium of the lightguide, is established.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 161–176, 1990.     

THE APPLICATION OF RANDOM MATRICES IN MATHEMATICAL PHYSICS

In this paper, we introduce the application of random matrices in mathematical physics including Riemann-Hilbert problem, nuclear physics, big data, image processing, compressed sensing and so on. We start with the Riemann-Hilbert problem and state the relation between the probability distribution of nontrivial zeros and the eigenvalues of the random matrices. Through the random matrices theory, we derive the distribution of Neutron width and probability density between energy levels. In addition, the application of random matrices in quantum chromo dynamics and two dimensional Einstein gravity equations is also present in this paper.