Pencils Of Matrices Research Articles

Asymptotic solutions of singularly perturbed linear differential-algebraic equations with periodic coefficients

The paper deals with the problem of constructing asymptotic solutions for singular perturbed linear differential-algebraic equations with periodic coefficients. The case of multiple roots of a characteristic equation is studied. It is assumed that the limit pencil of matrices of the system has one eigenvalue of multiplicity n, which corresponds to two finite elementary divisors and two infinite elementary divisors whose multiplicity is greater than 1.A technique for finding the asymptotic solutions is developed and n formal linearly independent solutions are constructed for the corresponding differential-algebraic system. The developed algorithm for constructing formal solutions of the system is a nontrivial generalization of the corresponding algorithm for constructing asymptotic solutions of a singularly perturbed system of differential equations in normal form, which was used in the case of simple roots of the characteristic equation.The modification of the algorithm is based on the equalization method in a special way the coefficients at powers of a small parameter in algebraic systems of equations, from which the coefficients of the formal expansions of the searched solution are found. Asymptotic estimates for the terms of these expansions with respect to a small parameter are also given.For an inhomogeneous differential-algebraic system of equations with periodic coefficients, existence and uniqueness theorems for a periodic solution satisfying some asymptotic estimate are proved, and an algorithm for constructing the corresponding formal solutions of the system is developed. Both critical and non-critical cases are considered.

The boundary value problems for the linear singularly perturbed systems of differential-algebraic equations

The article considers the main stages of the development of the theory of asymptotic integration of boundary-value problems for linear singularly perturbed differential-algebraic systems. The need of developing constructive methods of approximate integration of boundary-value problems for differential-algebraic systems is due to the importance of their practical application in the theory of nonlinear oscillations, stability of motion, control theory, radio engineering, and biology.
 In the present paper the authors offer a review of literary sources, which consider the methods of constructing asymptotic solutions of singularly perturbed systems of differential equations with a degenerate matrix with derivatives under the condition of stability of the spectrum of the limit pencil of matrices. It is noted that the problem of constructing asymptotic solutions of boundary-value problems for systems of this type is poorly studied, and therefore relevant. In particular, the question of the conditions for the existence and uniqueness of the solutions of these problems and the development of methods for constructing their asymptotics in various cases related to the behavior of the spectrum of the limit pencil of matrices has been poorly researched.

Generation of orthogonal rational functions by procedures for structured matrices

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

A block form of a singular pencil of operators and a method of obtaining it

A block form of a singular operator pencil $\lambda A+B$, where $\lambda$ is a complex parameter, and the linear operators $A$, $B$ act in finite-dimensional spaces, is described. An operator pencil $\lambda A+B$ is called regular if $n = m = rk(\lambda A+B)$, where $rk(\lambda A+B)$ is the rank of the pencil and $m$, $n$ are the dimensions of spaces (the operators map an $n$-dimensional space into an $m$-dimensional one); otherwise, if $n \ne m$ or $n = m$ and $rk(\lambda A+B)<n$, the pencil is called singular (irregular). The block form (structure) consists of a singular block, which is a purely singular pencil, i.e., it is impossible to separate out a regular block in this pencil, and a regular block. In these blocks, zero blocks and blocks, which are invertible operators, are separated out. A method of obtaining the block form of a singular operator pencil is described in detail for two special cases, when $rk(\lambda A+B) = m < n$ and $rk(\lambda A+B) = n < m$, and for the general case, when $rk(\lambda A+B) < n, m$. Methods for the construction of projectors onto subspaces from the direct decompositions, relative to which the pencil has the required block form, are given. Using these projectors, we can find the form of the blocks and, accordingly, the block form of the pencil. Examples of finding the block form for the various types of singular pencils are presented. To obtain the block form, in particular, the results regarding the reduction of a singular pencil of matrices to the canonical quasidiagonal form, which is called the Weierstrass-Kronecker canonical form, are used. Also, methods of linear algebra are used. The obtained block form of the pencil and the corresponding projectors can be used to solve various problems. In particular, it can be used to reduce a singular semilinear differential-operator equation to the equivalent system of purely differential and purely algebraic equations. This greatly simplifies the analysis and solution of differential-operator equations.

Stability of a Spline Collocation Difference Scheme for a Quasi-Linear Differential Algebraic System of First-Order Partial Differential Equations

A quasi-linear differential algebraic system of partial differential equations with a special structure of the pencil of Jacobian matrices of small index is considered. A nonlinear spline collocation difference scheme of high approximation order is constructed for the system by approximating the required solution by a spline of arbitrary in each independent variable. It is proved by the simple iteration method that the nonlinear difference scheme has a solution that is uniformly bounded in the grid space. Numerical results are demonstrated using a test example.

Matrix positivity preservers in fixed dimension. I

A classical theorem proved in 1942 by I.J. Schoenberg describes all real-valued functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size; such functions are necessarily analytic with non-negative Taylor coefficients. Despite the great deal of interest generated by this theorem, a characterization of functions preserving positivity for matrices of fixed dimension is not known.In this paper, we provide a complete description of polynomials of degree N that preserve positivity when applied entrywise to matrices of dimension N. This is the key step for us then to obtain negative lower bounds on the coefficients of analytic functions so that these functions preserve positivity in a prescribed dimension. The proof of the main technical inequality is representation theoretic, and employs the theory of Schur polynomials. Interpreted in the context of linear pencils of matrices, our main results provide a closed-form expression for the lowest critical value, revealing at the same time an unexpected spectral discontinuity phenomenon.Tight linear matrix inequalities for Hadamard powers of matrices and a sharp asymptotic bound for the matrix-cube problem involving Hadamard powers are obtained as applications. Positivity preservers are also naturally interpreted as solutions of a variational inequality involving generalized Rayleigh quotients. This optimization approach leads to a novel description of the simultaneous kernels of Hadamard powers, and a family of stratifications of the cone of positive semidefinite matrices.

Spectra of a class of non-self-adjoint matrices

We consider a new class of non-self-adjoint matrices that arise from an indefinite self-adjoint linear pencil of matrices, and obtain the spectral asymptotics of the spectra as the size of the matrices diverges to infinity. We prove that the spectrum is qualitatively different when a certain parameter c equals 0, and when it is non-zero, and that certain features of the spectrum depend on Diophantine properties of c.

To solving spectral problems for q-parameter polynomial matrices. III

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.

ON THE UNIVERSALITY OF THE DISTRIBUTION OF THE GENERALIZED EIGENVALUES OF A PENCIL OF HANKEL RANDOM MATRICES

Universality properties of the distribution of the generalized eigenvalues of a pencil of random Hankel matrices, arising in the solution of the exponential interpolation problem of a complex discrete stationary process, are proved under the assumption that every finite set of random variables of the process have a multivariate spherical distribution. An integral representation of the condensed density of the generalized eigenvalues is also derived. The asymptotic behavior of this function turns out to depend only on stationarity and not on the specific distribution of the process.

On the condensed density of the generalized eigenvalues of pencils of Gaussian random matrices and applications

Pencils of matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance are considered. An approximation to the distribution of the squared modulus of their determinant is computed which allows to get a closed form approximation of the condensed density of the generalized eigenvalues of the pencils. Implications of this result for solving several moments problems are discussed and some numerical examples are provided.

To solving inverse eigenvalue problems for matrices

The paper considers additive and multiplicative inverse eigenvalve problems and suggests a solution method based on the connection of inverse problems with the problem of computing the regular spectrum of a multiparameter pencil of constant matrices corresponding to the problem considered. Necessary and sufficient conditions for the solvability of inverse eigenvalve problems are established. Bibliography: 3 titles.

Construction of an asymptotic solution of one optimal control problem

We construct an asymptotic solution of a singularly perturbed optimal control problem in the case of a regular pencil of boundary matrices.

A Linear Programming Approach to Parameter Fitting for the Master Equation

This technical note proposes a new framework for the design of continuous time, finite state space Markov processes. In particular, we propose a paradigm for selecting an optimal matrix within a pre-specified pencil of transition rate matrices. Given any transition rate matrix specifying the time-evolution of the Markov process, we propose a class of figures of merit that upper-bounds the long-term evolution of any statistical moment. We show that optimization with respect to the aforementioned class of cost functions is tractable via dualization and linear programming methods. In addition, we suggest how this approach can be used as a tool for the sub-optimal design of the master equation, with performance guarantees. Our results are applied to illustrative examples.

Construction of an asymptotic solution of the Cauchy problem for a degenerate linear system in the singular case

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for a singularly perturbed linear system of differential equations with degenerate matrix of coefficients of derivatives in the case where the limit pencil of matrices is singular.

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.

Analytical Design of Invariant Controllers

Constructive conditions for the invariance of linear systems are derived from pencils of reduced-order matrices and applied to develop a method of design of invariant controllers. Examples illustrating the application of the analytical design method are given.

Methods for solving spectral problems for multiparameter matrix pencils

Methods for solving the partial eigenproblem for multiparameter regular pencils of real matrices, which allow one to improve given approximations of an eigenvector and the associated point of the spectrum (both finite and infinite) are suggested. Ways of extending the methods to complex matrices, polynomial matrices, and coupled multiparameter problems are indicated. Bibliography: 10 titles.

Several observations on symplectic, Hamiltonian, and skew-Hamiltonian matrices

We prove a Hamiltonian/skew-Hamiltonian version of the classical theorem relating strict equivalence and T-congruence between pencils of complex symmetric or skew-symmetric matrices. Then, we give a pure symplectic variant of the recent result of Xu concerning the singular value decomposition of a conjugate symplectic matrix. Finally, we discuss implications that can be derived from Veselić’s result on definite pairs of Hermitian matrices for the skew-Hamiltonian situation.

On the Construction of a Solution of a Degenerate Linear System of Differential Equations in the Neighborhood of an Irregular Singular Point

We investigate the asymptotics of the general solution of the linear system of differential equations with irregular singular point $$x^{ - h} B(x)\frac{{dy}}{{dx}} = A(x)y$$ in the case where the limit matrix coefficient of the derivative degenerates. Using the Newton diagram method, we construct the general solution of the system in the case where the regular pencil of matrices L(λ) = A0 − λB0 has multiple finite and infinite elementary divisors.

Commutative algebras of rational function matrices as endomorphisms of Kronecker modules I

Given a field K, a Kronecker module V is a pair of K-linear spaces ( U, V) together with a K-bilinear map K 2× U→ V. In finite dimensions this is also the notion of pencils of matrices. Every K[ X]-module can be construed as a Kronecker module. In particular the K[ X]-submodules of K( X) give rise to the Kronecker modules R h where h is a height function, i.e. a function h: K∪{∞}→{∞,0,1,2,…}. The K[ X]-module K[ X] itself gives the Kronecker module P that goes with the height function which is ∞ at ∞ and 0 on K. The modules R h that are infinite-dimensional come up precisely when h attains the value ∞ or when h is stictly positive on an infinite subset of K∪{∞}. The endomorphism algebra of R h is called a pole algebra. Those Kronecker modules V that are extensions of finite-dimensional submodules of P by infinite-dimensional R h lead to some engaging problems with matrix algebras. This is because the endomorphisms of such V constitute a K-subalgebra of n× n matrices over K( X), which is commutative if the extension is indecomposable. Among the algebras that are known to arise in the 2 × 2 case, when the extension is by P, are the coordinate rings of all elliptic curves. In this paper we replace P by an arbitrary infinite-dimensional R h . The following new algebras are realized: infinite-dimensional pole algebras End R h where h=0 on an infinite subset of K; maximal subalgberas of A× B for some pole algebras A, B; the quasi local ring K∝ S where S is a K-vector space of dimension at most card K. In the process we identify those height functions h that will tolerate an indecomposable extension V having non-trivial endomorphisms.