Spectral problems for matrix pencils. Methods and algorithms. I

A brief review of methods and algorithms to solve spectral problems for general-form linear matrix pencils is presented. The main attention is paid to solution of spectral problems for singular pencils. The methods are based on the idea to reduce the pencil dimension by using deflating subspaces and some types of matrix factorizations. Strictly equivalent transformations ensuring a reduction in pencil size are performed primarily by using elementary unitary matrices. Methods and algorithms to solve spectral problems are grouped according to the pencil type (regular or singular) and to the problem formulation (in computing scalar or vector spectral characteristics). INTRODUCTION A matrix pencil of integer degree s ̂ 1 is defined to be an expression of the form where Cf , ί = 1, . . . , n, are m χ η matrices. In the case of linear pencils (s = 1) the notation Ο(λ) = Α-λΒ is used. The pencil D(X) is said to be regular if m = n and the determinant det Ώ(λ) is not identically equal to zero. Otherwise, the pencil D(A) is said to be singular. Different spectral problems for matrix pencils are to be solved. (1) Computation of all the eigenvalues of a regular pencil or of a part of them, separation of the regular block of a singular pencil and subsequent computation of the eigenvalues of that block. (2) Determination of the structure of the JordanWeierstrass form of a regular linear pencil, i.e. of the elementary divisors of the pencil, corresponding to its finite and infinite eigenvalues. (3) Determination of the structure of the Kronecker form of a singular linear pencil, i.e. of the elementary divisors and minimal indices and of the polynomial solutions of the pencil. (4) Determination of the structure of the Smith form of a pencil of degree s ̂ 1, i.e. of the elementary divisors and of the null space dimensions. (5) Computation of the eigenvectors and of those vectors which form Jordan chains corresponding to the computed eigenvalues of a pencil of degree s ̂ 1. (6) Computation of minimal indices and of corresponding polynomial solutions which form a fundamental solution row of a pencil of degree s > 1. The solution of spectral problems for matrix pencils is required in many fields of science. We refer, for instance, to problems arising in stability analysis of frame constructions, to computation of zeros and poles of transfer functions in the theory of optimal control and the theory of electric circuits, to solution of general-type 338 V. B. Khazanov and V. N. Kublanovskaya differential equations including singular ones, and also to other mathematical models which are formulated directly in terms of matrix pencils. Among problems listed above the so-called generalized eigenvalue problem, i.e. the eigenvalue problem for a regular linear pencil is now the most important one for applications. To solve this problem numerically, a large number of algorithms have been developed which are widely presented in the literature (see, for example [10, 16, 40, 48, 49]), and many of them have been implemented on computers. The intensive investigation of spectral characteristics of general-form (singular) linear matrix pencils from the standpoint of numerical aspects started only in the last decade. This is also true for spectral characteristics of polynomial pencils of degree 5 > 1. The present paper 'Spectral problems for matrix pencils. Methods and algorithms' consists of three parts. In the first one given below, we consider those methods and algorithms to solve spectral problems for general-form linear pencils which are based on strictly equivalent pencil transformations and use mainly the techniques of elementary rotation and reflection matrices. Next parts to be published in a sequel to this paper consider those methods and algorithms which are applicable to solve spectral problems not only for linear matrix pencils but for polynomial pencils of degree s > 1 as well. These methods and algorithms are based, as a rule, on equivalent pencil transformations and use the techniques of elementary unimodular matrices. 1. THEORETICAL BACKGROUND Below, the simplest canonic forms of a linear matrix pencil are recalled and their structural characteristics are described. To pass from an arbitrary pencil A — λΒ to a very simple canonic form, strictly equivalent transformations are used. Pencils Ο^λ) and Ώ2(λ) of the same m χ η size are said to be strictly equivalent if they are interrelated via the formula Ώ^λ) = Ρί>2(Α)β, where Ρ and β are /l-independent non-singular matrices of order m and n, respectively. 1.1. Simplest canonic forms of a linear pencil The following theorems [15] hold. Theorem 1.1. An arbitrary regular pencil Ώ(λ) = Α — λΒ can be reduced by strictly equivalent transformations to the JordanWeierstrass canonic form = block diag { J λΙ,Ι λΝ}. Here, / is the identity matrix, J is a quasi-diagonal matrix with t x t Jordan blocks of the form