Symmetric Matrix Pencil Research Articles

Extension of Brauer and Rado perturbation theorems for regular matrix pencils

In this paper, we propose new results for changing eigenvalues of a regular matrix pencil A − λ B, which are based on the well-known Brauer’s theorem [A Brauer, Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 19, 75-91, 1952] and Rado’s theorem [B N Parlett, Symmetric matrix pencils, J. Comput. Appl. Math., 38, 373-385, 1991.]. These results allow us to change eigenvalues of the original matrix pencil without altering its regularity and in a quite simple way, even allowing to change infinite eigenvalues. We also present an extension of Rado’s theorem that allows changing eigenvalues of a regular symmetric matrix pencil without altering its symmetric structure, and we show how to use these results in order to change the eigenvalues of a quadratic polynomial matrix. Finally, we present numerical examples that confirm the expected results with the new extensions of these theorems.

The Numerical Solution of Large-Scale Generalized Eigenvalue Problems Arising from Finite-Element Modeling of Electroelastic Materials

The generalized eigenvalue problem for a symmetric definite matrix pencil obtained from finite-element modeling of electroelastic materials is solved numerically by the Lanczos algorithm. The mass matrix is singular in the considered problem, and therefore the process proceeds with the semi-inner product defined by this matrix. The shift-and-invert Lanczos algorithm is used to find multiple eigenvalues closest to some shift and the corresponding eigenvectors. The results of the numerical experiments are presented.

Backward error analysis and inverse eigenvalue problems for Hankel and Symmetric-Toeplitz structures

This work deals with the study of structured backward error analysis of Hankel and symmetric-Toeplitz matrix pencils. These structured matrix pencils belong to the class of symmetric matrix pencils with some additional properties that a symmetric matrix pencil does not have in general. The perturbation analysis of these two structures is discussed one by one to depict the additional properties explicitly. Present work shows the entrywise structured perturbation of matrix pencils in Frobenius norm such that the specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix pencil. The framework used here maintains the sparsity in the perturbation of the above-structured matrix pencils. Further, the backward error results help for solving a variety of inverse eigenvalue problems.

Inverse Numerical Range and Determinantal Quartic Curves

A hyperbolic ternary form, according to the Helton–Vinnikov theorem, admits a determinantal representation of a linear symmetric matrix pencil. A kernel vector function of the linear symmetric matrix pencil is a solution to the inverse numerical range problem of a matrix. We show that the kernel vector function associated to an irreducible hyperbolic elliptic curve is related to the elliptic group structure of the theta functions used in the Helton–Vinnikov theorem.

Generic symmetric matrix pencils with bounded rank

We show that the set of n \times n complex symmetric matrix pencils of rank at most r is the union of the closures of \lfloor r/2\rfloor +1 sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of n \times n complex symmetric matrix pencils of rank at most r . We also show that the irreducible components of the set of n\times n symmetric matrix pencils with rank at most r , when considered as an algebraic set, are among these closures.

A spectral Newton-Schur algorithm for the solution of symmetric generalized eigenvalue problems

This paper proposes a numerical algorithm based on spectral Schur complements to compute a few eigenvalues and the associated eigenvectors of symmetric matrix pencils. The proposed scheme follows an algebraic domain decomposition viewpoint and transforms the generalized eigenvalue problem into one of computing roots of scalar functions. These scalar functions are defined so that their roots are equal to the eigenvalues of the original pencil, and these roots are computed by Newton's method. We describe the theoretical aspects of the proposed scheme and demonstrate its performance on a few test problems.

Miniversal deformations of pairs of symmetric matrices under congruence

For each pair of complex symmetric matrices (A,B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices (A˜,B˜), close to (A,B) can be reduced by congruence transformation that smoothly depends on the entries of A˜ and B˜. Such a normal form is called a miniversal deformation of (A,B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A,B) to its miniversal deformation.

A GPU Solver for Sparse Generalized Eigenvalue Problems With Symmetric Complex-Valued Matrices Obtained Using Higher-Order FEM

This paper discusses a fast implementation of the stabilized locally optimal block preconditioned conjugate gradient method, using a hierarchical multilevel preconditioner to solve non-Hermitian sparse generalized eigenvalue problems with large symmetric complex-valued matrices obtained using the higher-order finite-element method (FEM), applied to the analysis of a microwave resonator. The resonant frequencies of the low-order modes are the eigenvalues of the smallest real part of a complex symmetric (though non-Hermitian) matrix pencil. These types of pencils arise in the FEM analysis of resonant cavities loaded with a lossy material. To accelerate the computations, graphics processing units (GPU, NVIDIA Pascal P100) were used. Single and dual-GPU variants are considered and a GPU-memory-saving implementation is proposed. An efficient sliced ELLR-T sparse matrix storage format was used and operations were performed on blocks of vectors for best performance on a GPU. As a result, significant speedups (exceeding a factor of six in some computational scenarios) were achieved over the reference parallel implementation using a multicore central processing unit (CPU, Intel Xeon E5-2680 v3, and 12 cores). These results indicate that the solution of generalized eigenproblems needs much more GPU memory than iterative techniques when solving a sparse system of equations, and also requires a second GPU to store some data structures in order to reduce the footprint, even for a moderately large systems.

Generic rank-one perturbations of structured regular matrix pencils

Classes of regular, structured matrix pencils are examined with respect to their spectral behavior under a certain type of structure-preserving rank-1 perturbations. The observed effects are as follows: On the one hand, generically the largest Jordan block at each eigenvalue gets destroyed or becomes of size one under a rank-1 perturbation, depending on that eigenvalue occurring in the perturbating pencil or not. On the other hand, certain Jordan blocks of T-alternating matrix pencils occur in pairs, so that in some cases, the largest block cannot just be destroyed or shrunk to size one without violating the pairing. Thus, the largest remaining Jordan block will typically increase in size by one in these cases. Finally, these results are shown to carry over to the classes of palindromic and symmetric matrix pencils.

Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations

The set of all solutions to the homogeneous system of matrix equations (XTA +AX,XTB + BX) = (0, 0), where (A,B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A,B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. K°agstr¨om, and V.V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375–3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.

Weighted non‐linear criterion‐based adaptive generalised eigendecomposition

Generalised eigendecomposition problem for a symmetric matrix pencil is reinterpreted as an unconstrained minimisation problem with a weighted non-linear criterion. The analytical results show that the proposed criterion has a unique global minimum which corresponds to the principal generalised eigenvectors, thus guaranteeing the global convergence via iterative methods to search the minimum. A gradient-based adaptive algorithm and a fixed point iteration-based adaptive algorithm are derived for the generalised eigendecomposition, which both work in parallel and avoid the error propagation effect of sequential-type algorithms. By applying the stochastic approximation theory, the global convergence of the proposed adaptive algorithm is proved. The performance of the proposed method is evaluated by simulations in terms of convergence rate, estimation accuracy as well as tracking capability.

A fully parallel method for the singular eigenvalue problem

In this paper, a fully parallel method for finding some or all finite eigenvalues of a real symmetric matrix pencil ( A, B) is presented, where A is a symmetric tridiagonal matrix and B is a diagonal matrix with b 1 > 0 and b i ≥ 0, i = 2,3,…, n. The method is based on the homotopy continuation with rank 2 perturbation. It is shown that there are exactly m disjoint, smooth homotopy paths connecting the trivial eigenvalues to the desired eigenvalues, where m is the number of finite eigenvalues of ( A, B). It is also shown that the homotopy curves are monotonic and easy to follow.

Real congruence of complex matrix pencils and complex projections of real Veronese varieties

Quadratically parametrized maps from a real projective space to a complex projective space are constructed as projections of the Veronese embedding. A classification theorem relates equivalence classes of projections to real congruence classes of complex symmetric matrix pencils. The images of some low-dimensional cases include certain quartic curves in the Riemann sphere, models of the real projective plane in complex projective 4-space, and some normal form varieties for real submanifolds of complex space with CR singularities.

Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils

The rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with different shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil approximates the solution of the original pencil. Different types of Ritz values and harmonic Ritz values are described and compared. Periodical purging of uninteresting directions reduces the size of the basis and makes it possible to compute many linearly independent eigenvectors and principal vectors of pencils with multiple eigenvalues. Relations to iterative methods are established. Results are reported for two large test examples. One is a symmetric pencil coming from a finite element approximation of a membrane; the other is a nonsymmetric matrix modeling an idealized aircraft stability problem.

Computation of Derivatives of Repeated Eigenvalues and the Corresponding Eigenvectors of Symmetric Matrix Pencils

This paper presents and analyzes new algorithms for computing the numerical values of derivatives, of arbitrary order, and of eigenvalues and eigenvectors of ${\bf A}(\rho){\bf x}(\rho) = \lambda(\rho){\bf B}(\rho){\bf x}(\rho)$ at a point $\rho=\rho_0$ at which the eigenvalues considered are multiple. Here ${\bf A}(\rho)$ and ${\bf B}(\rho)$ are hermitian matrices which depend analytically on a single real variable $\rho,$ and ${\bf B}(\rho_0)$ is positive definite. The algorithms are valid under more general conditions than previous algorithms. Numerical results support the theoretical analysis and show that the algorithms are also useful when eigenvalues are merely very close rather than coincident.

Trace minimization and definiteness of symmetric pencils

A symmetric matrix pencil A - λB of order n is called positive definite if there is a μ such that the matrix A − μB is positive definite. We consider the case with B nonsingular and show that the definiteness is closely related to the existence of min Tr X T AX under the condition X T BX = J 1 where J 1 is a given diagonal matrix of order ≤ n and J 2 1 = I. We also prove an analog of the Cauchy interlacing theorem for some such pencils.

Computing the inertias in symmetric matrix pencils

We define and study generalized pencil eigenvalues for a pencil P( S, T) of real symmetric matrices S and T. This allows us to find all occurring inertias among pencil members aS + bT theoretically and numerically. Computed examples based on a MATLAB code are included.

Rayleigh-Ritz and Lanczos methods for symmetric matrix pencils

We are concerned with eigenvalue problems for definite and indefinite symmetric matrix pencils. First, Rayleigh-Ritz methods are formulated and, using Krylov subspaces, a convergence analysis is presented for definite pencils. Second, generalized symmetric Lanczos algorithms are introduced as a special Rayleigh-Ritz method. In particular, an a posteriori convergence criterion is demonstrated by using residuals. Local convergence to real and nonreal eigenvalues is also discussed. Numerical examples concerning vibrations of damped cantilever beams are included.

A homotopy algorithm for a symmetric generalized eigenproblem

In this paper, a homotopy algorithm for finding all eigenpairs of a real symmetric matrix pencil (A, B) is presented, whereA andB are realn×n symmetric matrices andB is a positive semidefinite matrix. In the algorithm, pencil (A, B) is first reduced to a pencil\((\tilde A,\tilde B)\), where\(\tilde A\) is a symmetric tridiagonal matrix and\(\tilde B\) is a positive definite and diagonal matrix. Then, the “Divide and Conquer” strategy with homotopy continuation approach is used to find all eigenpairs of pencil\((\tilde A,\tilde B)\). One can easily form the eigenpair (x,λ) of pencil (A, B) from the eigenpair (y, λ) of pencil\((\tilde A,\tilde B)\) with a few computations. Numerical comparisons of our algorithm with the QZ algorithm in the widely used EISPACK library are presented. Numerical results show that our algorithm is strongly competitive in terms of speed, accuracy and orthogonality. The performance of the parallel version of our algorithm is also presented.

Variational and numerical methods for symmetric matrix pencils

A review is presented of some recent advances in variational and numerical methods for symmetric matrix pencils λA – B in which A is nonsingular, A and B are hermitian, but neither is definite. The topics covered include minimax and maximin characterisations of eigenvalues, perturbation by semidefinite matrices and interlacing properties of real eigenvalues, Rayleigh quotient algorithms and their convergence properties, Rayleigh-Ritz methods employing Krylov subspaces, and a generalised Lanczos algorithm.