Perturbation and robust stability of autonomous singular linear matrix difference equations

In the perturbation theory of linear matrix difference equations, it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear. In this paper, the perturbation theory of complex Weierstrass canonical form for regular matrix pencils is investigated. By using matrix pencil theory and the Weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil. Moreover we study robust stability of a class of linear matrix difference equations (of first and higher order) whose coefficients are square constant matrices.     

Pencils with prescribed constant subpencils

The results in this paper are within the scope of matrix pencil completion problems. Given an arbitrary matrix pencil, we obtain necessary and sufficient conditions for the existence of a strictly equivalent pencil with a prescribed constant subpencil for algebraically closed fields.     

An iterative method for solving the stable subspace of a matrix pencil and its application

This work is to propose an iterative method of choice to compute a stable subspace of a regular matrix pencil. This approach is to define a sequence of matrix pencils via particular left null spaces. We show that this iteration preserves a semigroup property depending only on the initial matrix pencil. Via this recursion relationship, we propose an accelerated iterative method to compute the stable subspace and use it to provide a theoretical result to solve the principal square root of a given matrix, both nonsingular and singular. We show that this method can not only find out the matrix square root, but also construct an iterative approach which converges to the square root with any desired order.     

A deflation method for regular matrix pencils

A generalization of the concept of eigenvalue is introduced for a matrix pencil and it is called eigenpencil; an eigenpencil is a pencil itself and it contains part of the spectral information of the matrix pencil. A Wielandt type deflation procedure for regular matrix pencils is developed, using eigenpencils and supposing that they can have both finite and infinite eigenvalues. A numerical example illustrates the proposed method.     

The Spectral Variation of Pencils of Matrices

Perturbation theorems for the spectrum of a regular matrix pencil $λA-B$ are given. As it may include points near or at infinity the Euclidean distance is not appropriate. We use the chordal metric and the distances. For those purpose we develop here an algebraic treatment of matrix pairs, with special reference to diagonable and definite pairs, using ideas from the theory of matrix polynomials.     

The canonical structure of a pencil of degenerate matrix functions

In this paper we study global properties of a pencil of identically degenerate matrix functions with a compact domain of definition. Matrix functions are assumed to have a constant rank and all roots of the characteristic equation of the matrix pencil are assumed to have a constant multiplicity at each point in the domain of definition. We obtain sufficient conditions for the smooth orthogonal similarity of matrix functions to the upper triangular form and sufficient conditions for the smooth equivalence of the pencil of matrix functions to its canonical form. We illustrate the obtained results with simple examples.     

Systèmes hyperboliques d'équations aux dérivées partielles linéaires : régularité L2loc et matrices diagonalisables

The well-posedness of a Cauchy problem issue from an hyperbolic linear system is linked to the spectral properties of a real matrix pencil. It is known that such a problem is well posed in L² if and only if the imaginary exponential of the pencil is bounded. We give a condition to have a bounded exponential when the eigenvalues don't have the same multiplicities. For pencils spanned by two 3×3 matrices, we prove that the exponential is bounded if and only if the pencil is analytically diagonable.     

Spectral factorization of non-symmetric polynomial matrices

The topic of the paper is spectral factorization of rectangular and possibly non-full-rank polynomial matrices. To each polynomial matrix we associate a matrix pencil by direct assignment of the coefficients. The associated matrix pencil has its finite generalized eigenvalues equal to the zeros of the polynomial matrix. The matrix dimensions of the pencil we obtain by solving an integer linear programming (ILP) minimization problem. Then by extracting a deflating subspace of the pencil we come to the required spectral factorization. We apply the algorithm to most general-case of inner–outer factorization, regardless continuous or discrete time case, and to finding the greatest common divisor of polynomial matrices.     

广义线性系统平衡点的谱分析

从另一个角度研究线性广义系统Ex＝Ax的平衡点邻域的定性行为,并对正则性概念作了有益的推广.     

Spectral problem for polynomial matrix pencils. 2

For an arbitrary polynomial pencil of matrices A_i of dimensions m×n one presents an algorithm for the computation of the eigenvalues of the regular kernel of the pencil. The algorithm allows to construct a regular pencil having the same eigenvalues as the regular kernel of the initial pencil or (in the case of a dead end termination) allows to pass from the initial pencil to a pencil of smaller dimensions whose regular kernel has the same eigenvalues as the initial pencil. The problem is solved by reducing the obtained pencil to a linear one. For solving the problem in the case of a linear pencil one considers algorithms for pencils of full column rank as well as for completely singular pencils.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 109–116, 1981.     

Construction of a canonic basis for matrices and pencils of matrices

An algorithm is offered, which with insignificant modifications permits; 1) the finding of a canonic basis of the root sub space corresponding to a prescribed eigenvalue of a matrix; 2) the finding of chains of associated vectors to the eigenvectors corresponding to a prescribed eigenvalue of a regular linear pencil; 3) the finding of chains of generalized associated vectors corresponding to a prescribed eigenvalue of a regular kernel of a singular linear pencil of complete column rank of two matrices; 4) the finding of linearly independent polynomial solutions of a singular linear pencil. The algorithm consists in the construction of a finite sequence of certain auxiliary matrices the choice of which depends on the problem being solved and in the construction of a sequence of their null-spaces, enabling the obtaining of all necessary information on the unknown vectors of the canonic basis of the problem being solved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 46–62, 1979.     

Stability Issues in Regular and Noncritical Singular DAEs

This paper addresses equilibrium stability issues in both regular and singular differential-algebraic equations (DAEs). We present a survey of available results and discuss some commonly-used methods in the qualitative analysis of low-index autonomous systems. Additionally, we extend the use of matrix pencil theory to the stability study of singular problems, pointing out some interesting relations between regular and singular DAEs. This framework is applied to the qualitative analysis of singular equations arising in the context of the Singularity Induced Bifurcation theorem, and also to the stability study of stationary equilibria in singular DAEs.     

Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint

We consider minimization of a quadratic objective function subject to a sign-indefinite quadratic equality constraint. We derive necessary and sufficient conditions for the existence of solutions to the constrained minimization problem. These conditions involve a generalized eigenvalue of the matrix pencil consisting of a symmetric positive-semidefinite matrix and a symmetric indefinite matrix. A complete characterization of the solution set to the constrained minimization problem in terms of the eigenspace of the matrix pencil is provided.     

On Periodic Solutions of Degenerate Singularly Perturbed Linear Systems with Multiple Elementary Divisor

We establish sufficient conditions for the existence and uniqueness of a periodic solution of a system of linear differential equations with a small parameter and a degenerate matrix of coefficients of derivatives in the case of a multiple spectrum of a boundary matrix pencil. We construct asymptotics of this solution.     

The computation of Kronecker's canonical form of a singular pencil

We develop stable algorithms for the computation of the Kronecker structure of an arbitrary pencil. This problem can be viewed as a generalization of the well-known eigenvalue problem of pencils of the type λI?A. We first show that the elementary divisors (λ ? α)ⁱ of a regular pencil λB?A can be retrieved with a deflation algorithm acting on the expansion (λ ? α)B ? (A ? αB). This method is a straightforward generalization of Kublanovskaya's algorithm for the determination of the Jordan structure of a constant matrix. We also show how to use this method to determine the structure of the infinite elementary divisors of λB?A. In the case of singular pencils, the occurrence of Kronecker indices—containing the singularity of the pencil—somewhat complicates the problem. Yet our algorithm retrieves these indices with no additional effort, when determining the elementary divisors of the pencil. The present ideas can also be used to separate from an arbitrary pencil a smaller regular pencil containing only the finite elementary divisors of the original one. This is shown to be an effective tool when used together with the QZ algorithm.     

Operator pencils on the algebra of densities

We continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role played by the geometry of the extended manifold where the algebra of densities is a special class of functions. Firstly we consider basic examples. We give a projective line of diff(M)-equivariant pencil liftings for first order operators and describe the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO) pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO pencil lifting to describe all regular proj-equivariant pencil liftings. In particular, the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian.     

An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Summary. We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix , or a pair of left and right deflating subspaces of a regular matrix pencil . This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm. Received September 20, 1994 / Revised version received February 5, 1996     

On the simultaneous tridiagonalization of two symmetric matrices

We discuss congruence transformations aimed at simultaneously reducing a pair of symmetric matrices to tridiagonal–tridiagonal form under the very mild assumption that the matrix pencil is regular. We outline the general principles and propose a unified framework for the problem. This allows us to gain new insights, leading to an economical approach that only uses Gauss transformations and orthogonal Householder transformations. Numerical experiments show that the approach is numerically robust and competitive.