Jordan Chains Research Articles

An algorithm for computing Jordan chains and inverting analytic matrix functions

We present an efficient algorithm for obtaining a canonical system of Jordan chains for an n × n regular analytic matrix function A( λ) that is singular at the origin. For any analytic vector function b( λ), we show that each term in the Laurent expansion of A( λ) −1 b( λ) may be obtained from the previous terms by solving an ( n + d) × ( n+ d) linear system, where d is the order of the zero of det A( λ) at λ = 0. The matrix representing this linear system contains A(0) as a principal submatrix, which can be useful if A(0) is sparse. The last several iterations can be eliminated if left Jordan chains are computed in addition to right Jordan chains. The performance of the algorithm in floating point and exact (rational) arithmetic is reported for several test cases. The method is shown to be forward stable in floating point arithmetic.

The inverse problem for Krein orthogonal matrix functions

In the mid-fifties, in a seminal paper, M. G. Krein introduced continuous analogs of Szegő orthogonal polynomials on the unit circle and established their main properties. In this paper, we generalize these results and subsequent results that he obtained jointly with Langer to the case of matrix-valued functions. Our main theorems are much more involved than their scalar counterparts. They contain new conditions based on Jordan chains and root functions. The proofs require new techniques based on recent results in the theory of continuous analogs of resultant and Bezout matrices and solutions of certain equations in entire matrix functions.

Canonical structures for palindromic matrix polynomials

Spectral properties and canonical structures of palindromic matrix polynomials are studied in terms of their linearizations, standard triples, and unitary triples. These triples describe matrix polynomials via eigenvalues and Jordan chains. As an application of canonical structures and their properties, criteria are developed for stable boundedness of solutions of systems of linear differential equations with symmetries.

Pseudoperturbation method for computation of E. Schmidt eigenvalues

AbstractIn two problems with approximately given n ‐multiple generalized E. Schmidt eigenvalue with relevant Jordan chains pseudoperturbation method is applied for their sharpening. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

STURM–LIOUVILLE PROBLEMS WITH REDUCIBLE BOUNDARY CONDITIONS

AbstractThe regular Sturm–Liouville problem$$ \tau y:=-y''+qy=\lambda y\quad\text{on }[0,1],\ \lambda\in\CC, $$is studied subject to boundary conditions$$ P_j(\lambda)y'(j)=Q_j(\lambda)y(j),\quad j=0,1, $$where $q\in L^1(0,1)$ and $P_j$ and $Q_j$ are polynomials with real coefficients. A comparison is made between this problem and the corresponding ‘reduced’ one where all common factors are removed from the boundary conditions. Topics treated include Jordan chain structure, eigenvalue asymptotics and eigenfunction oscillation.

The resultant approach to computing vector characteristics of multiparameter polynomial matrices

Known types of resultant matrices corresponding to one-parameter matrix polynomials are generalized to the multiparameter case. Based on the resultant approach suggested, methods for solving the following problems for multiparameter polynomial matrices are developed: computing a basis of the matrix range, computing a minimal basis of the right null-space, and constructing the Jordan chains and semilattices of vectors associated with a multiple spectrum point. In solving these problems, the original polynomial matrix is not transformed. Methods for solving other parametric problems of algebra can be developed on the basis of the method for computing a minimal basis of the null-space of a polynomial matrix. Issues concerning the optimality of computing the null-spaces of sparse resultant matrices and numerical precision are not considered. Bibliography: 19 titles.

The boundary layer phenomena in two‐dimensional transversely isotropic piezoelectric media by exact symplectic expansion

AbstractA separable variable method is introduced to find the exact homogeneous solutions of a two‐dimensional transversely isotropic piezoelectric media to handle general boundary conditions. The usual method of separable variables for partial differentiation equations cannot be readily applicable due to the tangling of the unknowns and their derivatives. Introducing dual variables of stresses, we obtain a set of first‐order Hamiltonian equations whose eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions. The solutions consist of two parts. The first part is the derogative zero‐eigenvalue solutions of the Saint Venant type together with all their Jordan chains. The second part is the decaying non‐zero‐eigenvalue solutions describing the boundary layer effects. The classical solutions are actually the zero‐eigenvalue solutions representing the simple extension, bending, equipotential field, and the uniform electric displacement. On the other hand, the non‐zero‐eigenvalue solutions represent the localized solutions, which are sensitive to the boundary conditions and are decaying rapidly with respect to the distance from the boundaries. Some rate‐of‐decay curves of the newly found non‐zero‐eigenvalue solutions are shown by numerical examples. Finally, the complete boundary layer effects are quantified for the first time. Copyright © 2006 John Wiley & Sons, Ltd.

Fundamental Operator-Functions of Degenerate Differential and Difference-Differential Operators in Banach Spaces Which Have a Noether Operator in the Principal Part

Continuing the study by the first author, we construct fundamental operator-functions for some classes of degenerate differential operators (both ordinary and partial) and difference-differential operators in Banach spaces which have a Noether operator at the leading (time) derivative. To this end, we use the method of Jordan chains of Noether operators and the methods of distribution theory in Banach spaces.

Laurent expansion for the determinant of the matrix of scalar resolvents

Let be an arbitrary square matrix, an eigenvalue of it, and two systems of linearly independent vectors. A representation of the matrix of scalar resolvents, with th entry equal by definition to , in the form of the product of three matrices , and is obtained, only one of which, , depends on and is a rational function of . On the basis of this factorization and the Binet-Cauchy formula a method for finding the principal part of the Laurent series at the point for the determinant of the matrix of scalar resolvents is put forward and the first two coefficients of the series are found. In the case when at least one of them is distinct from zero, the change after the transition from to of the part of the Jordan normal form corresponding to is determined, where is the operator of rank associated with the systems of vectors and ; and the Jordan basis for the corresponding root subspace of is constructed from Jordan chains of .

Natural Representations of the Multiplicity of an Analytic Operator-valued Function at an Isolated Point of the Spectrum

Representations are given for the multiplicity of an analytic operator-valued function A at an isolated point z0 of the spectrum in the form of kernels and ranges of Hankel and Toeplitz matrices whose entries are derived from the Taylor coefficients of A and the Laurent coefficients of A−1 about z0. In two special cases the results can be expressed in terms of finite matrices: when A is a polynomial and when A−1 has a pole at z0. The latter case leads to the theory of Jordan chains.

Pole cancellation

The problem of eliminating the right half plane poles of an rmvf (rational matrix valued function) G( z) with minimal realization G( z) = D + C( zI n − A) −1 B by multiplication on the right by a suitably chosen J-inner rmvf Θ( z) is considered from a number of different points of view, including the notion of minimal J conjugators that was introduced by Kimura, the null/pole structure of rmvf’s that is developed at length in the monograph of Ball–Gohberg–Rodman, and the theory of reproducing kernel Hilbert spaces. Connections between these different points of view are developed and correspondences between (1) the Jordan chains corresponding to the right half plane eigenvalues of A*, (2) the left null chains of Θ( z) in the sense of Ball–Gohberg–Rodman, and (3) the invariant subspaces of the generalized backwards shift operator applied to a suitably defined space of rmvf’s are established. Enroute, a theorem of Kimura that relates the existence of minimal pole conjugators to the existence of solutions of a related Riccati equation is refined and made more precise with the aid of the techniques referred to above.

On the change of the Jordan form under the transition from the adjacency matrix of a vertex-transitive digraph to its principal submatrix of co-order one

Let J( λ; n 1, …, n k ) be the set of matrices A such that λ is an eigenvalue of A and n 1 ⩽ ⋯ ⩽ n k are the sizes of the Jordan blocks associated with λ. For a given index v of A, denote by A − v the principal submatrix of co-order one obtained from A by deleting the vth row and column. In the present paper, all possible changes of the part of the Jordan form corresponding to λ under the transition from A to A − v are determined for matrices A ∈ J( λ; n 1, …, n k ) such that for the eigenvalue λ of both A and A ⊤, there exists a Jordan chain of the largest length n k whose eigenvector has nonzero vth entry. In particular, it is shown that for almost every matrix A ∈ J( λ; n 1, …, n k ), n 1, …, n k−1 are the sizes of Jordan blocks for λ considered as an eigenvalue of A − v. Moreover, it is also proved that if A is the adjacency matrix of a vertex-transitive digraph and k ⩾ 2, then the change n 1, …, n k → n 1, …, n k−2 , 2 n k−1 − 1 holds for the eigenvalue λ under the transition from A to A − v. In the case of k = 1, λ is a simple eigenvalue of A and does not belong to the spectrum of A − v.

On the lengths of Jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial

We describe a connection between the geometrical structure of the numerical range W ( L ) of a selfadjoint quadratic operator polynomial L ( λ ) and the lengths of Jordan chains for eigenvalues on the boundary of the numerical range. Also for each pair of natural numbers m , n (1⩽ m ⩽2 n ) we construct a monic quadratic polynomial L m , n ( λ )= λ 2 + λC + B with selfadjoint n × n matrices C , B which has an eigenvalue λ 0 ∈∂ W ( L m , n ) with a Jordan chain of length m .

On the computation of the Jordan canonical form of regular matrix polynomials

In this paper, an algorithm for the computation of the Jordan canonical form of regular matrix polynomials is proposed. The new method contains rank conditions of suitably defined block Toeplitz matrices and it does not require the computation of the Jordan chains or the Smith form. The Segré and Weyr characteristics are also considered.

On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank

We show that the r largest Jordan blocks disappear and all other blocks remain the same in the part of the Jordan form corresponding to a given eigenvalue λ under a generic rank r perturbation. Moreover, a necessary and sufficient condition on the entries of a perturbation under which the spectral properties of λ change in this manner is obtained with the use of the resolvent technique for the case in which the geometric multiplicity of λ is greater than or equal to r. A Jordan basis in the corresponding root space is constructed from the Jordan chains of the original matrix. A complete description of how the spectrum changes in a small neighborhood of the point z=λ is given for the case of a small parameter multiplying the perturbation.

On Regularization of Pseudoperturbation Method for Sharpening of Approximately Given Generalized Jordan Chains

AbstractCombination of pseudoperturbation and Newton‐Kantorovich (N.‐K.) methods are applied for the sharpening of approximately given eigenvalue and generalized Jordan chains (GJChs) of linear by spectral parameter operator function.

Degeneracy of Resonances, Jordan Blocks, and Gamow–Jordan Eigenfunctions

We show that a degeneracy of resonances is associated with a second rank pole in the scattering matrix and a Jordan chain of generalized eigenfunctions of the radial Schrodinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in complex resonance energy eigenfunctions. In this biorthonormal basis, any operator f(H r \(\overline B\) which is a regular function of the Hamiltonian is represented by a nondiagonal complex matrix with a Jordan block of rank 2.

On the computation of the Jordan canonical form of regular matrix polynomials

In this paper, an algorithm for the computation of the Jordan canonical form of regular matrix polynomials is proposed. The new method contains rank conditions of suitably defined block Toeplitz matrices and it does not require the computation of the Jordan chains or the Smith form. The Segré and Weyr characteristics are also considered.

On the lengths of Jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial

We describe a connection between the geometrical structure of the numerical range W( L) of a selfadjoint quadratic operator polynomial L( λ) and the lengths of Jordan chains for eigenvalues on the boundary of the numerical range. Also for each pair of natural numbers m, n (1⩽ m⩽2 n) we construct a monic quadratic polynomial L m, n ( λ)= λ 2+ λC+ B with selfadjoint n× n matrices C, B which has an eigenvalue λ 0∈∂ W( L m, n ) with a Jordan chain of length m.

Jordan blocks and Gamow-Jordan eigenfunctions associated with a degeneracy of unbound states

An accidental degeneracy of unbound states gives rise to a double pole in the scattering matrix, a double zero in the Jost function, and a Jordan chain of length 2 of generalized Gamow-Jordan eigenfunctions of the radial Schr\"odinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in bound- and resonant-state energy eigenfunctions plus a continuum of scattering wave functions of a complex wave number. In this biorthonormal basis, any operator ${f(H}_{r}^{(\mathcal{l})})$ which is a regular function of the Hamiltonian is represented by a complex matrix that is diagonal except for a Jordan block of rank 2. The occurrence of a double pole in the Green's function, as well as the non exponential time evolution of the Gamow-Jordan generalized eigenfunctions are associated with the Jordan block in the complex energy representation.