Eigenvalue Problem For Symmetric Matrices Research Articles

The inverse eigenvalue problem for symmetric anti-bidiagonal matrices

The inverse eigenvalue problem for real symmetric matrices of the form 0 0 0 ⋯ 0 0 ∗ 0 0 0 ⋯ 0 ∗ ∗ 0 0 0 ⋯ ∗ ∗ 0 · ⋮ ⋮ ⋮ · ⋮ ⋮ ⋮ · 0 0 ∗ ⋯ 0 0 0 0 ∗ ∗ ⋯ 0 0 0 ∗ ∗ 0 ⋯ 0 0 0 is solved. The solution is shown to be unique. The problem is also shown to be equivalent to the inverse eigenvalue problem for a certain subclass of Jacobi matrices.

On Some Inverse Eigenvalue Problems with Toeplitz-Related Structure

Some inverse eigenvalue problems for matrices with Toeplitz-related structure are considered in this paper. In particular, the solutions of the inverse eigenvalue problems for Toeplitz-plus-Hankel matrices and for Toeplitz matrices having all double eigenvalues are characterized, respectively, in close form. Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric Toeplitz matrices. Numerical testing results show a clear advantage of such an application.

On two conjectures regarding an inverse eigenvalue problem for acyclic symmetric matrices

For a given acyclic graph G, an important problem is to characterize all of the eigenbalues over all symmetric matrices with graph G. Of particular interest is the connection between this standard inverse eigenvalue problem and describing all the possible associated ordered multiplicity lists, along with determining the minimum number of distinct eigenvalues for a symmetric matrix with graph G. In this note two important open quesitons along these lines are resolved, both in the negative.&nbsp

Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

We characterize the possible lists of ordered multiplicities among matrices whose graph is a generalized star (a tree in which at most one vertex has degree greater than 2) or a double generalized star. Here, the inverse eigenvalue problem (IEP) for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with a conjecture that determination of the possible ordered multiplicities is equivalent to the IEP for a given tree. Moreover, a key spectral feature of the IEP in the case of generalized stars is shown to characterize them among trees.

EIGENVALUE PROBLEMS FOR VIBRATING STRUCTURES COUPLED WITH QUIESCENT FLUIDS WITH FREE SURFACE

Vibrations of plates, shells and plate–shell systems coupled with sloshing, quiescent and inviscid fluid have been advantageously studied by inserting the sloshing condition into the eigenvalue problem. Here a formulation of this particular eigenvalue problem for symmetric matrices is obtained. In fact, in the previous studies, this technique has given eigenvalue problems for non-symmetric matrices for which the problem of the existence of complex eigenvalues arises. The present analysis deals with compressible and incompressible fluids and the discretization of the system is obtained by using the Rayleigh–Ritz method. The Rayleigh quotient of the system is manipulated to obtain expressions suitable for symmetric formulations of the eigenvalue problem. In particular, the Rayleigh quotient is transformed into a simpler expression where the potential energies of the compressible fluid and free surface waves do not appear. The method is applied to a vertical, simply supported, circular cylindrical shell partially filled by an incompressible sloshing liquid. A case with large interaction between sloshing and bulging modes is considered and interesting phenomena are observed.

The generalized eigenvalue problem for tridiagonal symmetric interval matrices

An algorithm to compute the eigenintervals of the generalized eigenvalue problem for tridiagonal symmetric matrices (A,B) with interval entries is developed. The algorithm is based on the Sturm Sequences and requires twice as much computational effort as for point matrices. The algorithm is applied to the problem of computing the natural frequencies of a vibrating beam where the manufacturing tolerance is taken into consideration.

Classroom Note:Centrosymmetric Matrices

Attention is drawn to some earlier work on the odd--even properties of eigenvectors of centrosymmetric matrices.

The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application

We consider an infinite complex symmetric (not necessarily Hermitian) tridiagonal matrix T whose diagonal elements diverge to ∞ in modulus and whose off-diagonal elements are bounded. We regard T as a linear operator mapping a maximal domain in the Hilbert space l 2 into l 2. Assuming the existence of T −1 we consider the problem of approximating a given simple eigenvalue λ of T by an eigenvalue λ n of T n , and nth order principal submatrix of T. Let x = [ x (1), x (2),…] T be an eigenvector corresponding to λ. Assuming x T x ≠ 0 and f n+1x (n+1) x (n) → 0 as n → ∞, we show that there exists a sequence { λ n } of T n such that λ − λ n = f n + 1x (n+1)[1 + o(1)] (x Tx) → 0 , where f n+1 represents the ( n, n + 1) element of T. Application to the following problems is included: (a) solve J v ( z) = 0 for v, given z ≠ 0, and (b) compute the eigenvalues of the Mathieu equation. Fortunately, the existence of T −1 need not be verified for these examples since we may show that T + αI with α taken appropriately has an inverse.

Solving inverse eigenvalue problems by a projected newton method

The inverse eigenvalue problem for symmetric matrices (IEP) is formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. an iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method converges locally y quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal.,24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.

The Laguerre Iteration in Solving the Symmetric Tridiagonal Eigenproblem, Revisited

This paper presents an algorithm for the eigenvalue problem of symmetric tridiagonal matrices. The algorithm employs the determinant evaluation, split-and-merge strategy, and the Laguerre iteration. The method directly evaluates eigenvalues and uses inverse iteration as an option when eigenvectors are needed. This algorithm combines the advantages of existing algorithms such as QR, bisection/multisection, and Cuppen’s divide-and-conquer method. It is fully parallel and competitive in speed with the most efficient QR algorithm in serial mode. On the other hand, the algorithm is as accurate as any standard algorithm for the symmetric tridiagonal eigenproblem and enjoys the flexibility in evaluating partial spectrum.

The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J0( z) − iJ1( z) and of Bessel functions Jm( z) of any real order m

Consider computing simple eigenvalues of a given compact infinite matrix re- garded as operating in the complex Hilbert space l 2 by computing the eigenvalues of the truncated finite matrices and taking an obvious limit. In this paper we deal with a special case where the given matrix is compact, complex, and symmetric (but not necessarily Hermitian). Two examples of application are studied. The first is con- cerned with the equation J 0( z) − iJ 1( z)=0 appearing in the analysis of the solitary-wave runup on a sloping beach, and the second with the zeros of the Bessel function J m ( z) of any real order m. In each case, the problem is reformulated as an eigenvalue problem for a compact complex symmetric tridiagonal matrix operator in l 2 whose eigenvalues are all simple. A complete error analysis for the numerical solution by truncation is given, based on the general theorems proved in this paper, where the usefulness of the seldom used generalized Rayleigh quotient is demonstrated.

On a Multivariate Eigenvalue Problem, Part I: Algebraic Theory and a Power Method

Multivariate eigenvalue problems for symmetric and positive definite matrices arise from multivariate statistics theory where coefficients are to be determined so that the resulting linear combinations of sets of random variables are maximally correlated. By using the method of Lagrange multipliers such an optimization problem can be reduced to the multivariate eigenvalue problem. For over 30 years an iterative method proposed by Horst [Psychometrika, 26 (1961), pp. 129–149J has been used for solving the multivariate eigenvalue problem. Yet the theory of convergence has never been complete. The number of solutions to the multivariate eigenvalue problem also remains unknown. This paper contains two new results. By using the degree theory, a closed form on the cardinality of solutions for the multivariate eigenvalue problem is first proved. A convergence property of Horst’s method by forming it as a generalization of the so-called power method is then proved. The discussion leads to new formulations of numerical methods.

Construction of polynomials that are orthogonal with respect to a discrete bilinear form

We describe a new algorithm for the computation of recursion coefficients of monic polynomials {p j } =0/ that are orthogonal with respect to a discrete bilinear form (f, g) := Σ =1/ f(x k )g(x k )w k ,m ≥n, with real distinct nodesx k and real nonvanishing weightsw k . The algorithm proceeds by applying a judiciously chosen sequence of real or complex Givens rotations to the diagonal matrix diag[x 1,x 2, ...,x m ] in order to determine an orthogonally similar complex symmetric tridiagonal matrixT, from whose entries the recursion coefficients of the monic orthogonal polynomials can easily be computed. Fourier coefficients of given functions can conveniently be computed simultaneously with the recursion coefficients. Our scheme generalizes methods by Elhay et al. [6] based on Givens rotations for updating and downdating polynomials that are orthogonal with respect to a discrete inner product. Our scheme also extends an algorithm for the solution of an inverse eigenvalue problem for real symmetric tridiagonal matrices proposed by Rutishauser [20], Gragg and Harrod [17], and a method for generating orthogonal polynomials based theoron [18]. Computed examples that compare our algorithm with the Stieltjes procedure show the former to generally yield higher accuracy except whenn ≪m. Ifn is sufficiently much smaller thanm, then both the Stieltjes procedure and our algorithm yield accurate results.

Divide and conquer algorithms for the bandsymmetric eigenvalue problem

Divide and conquer algorithms are formulated for the solution of the eigenvalue problem for symmetric band matrices. The new algorithms are compared to the traditional solutionspaths offered by Eispack, tridiagonalization of the band matrix followed by the tridiagonal QR algorithm.

Inverse Eigenvalue Problems for Symmetric Toeplitz Matrices

The inverse eigenvalue problems for symmetric Toeplitz matrices with complex-valued (IEPSCTM) and real-valued (IEPSRTM) entries are studied. The main tools are complex and real algebraic geometry. In the complex case it is shown that the IEPSCTM is solvable for most spectra and always solvable for $n\leqq 4$. In the real case the natural decomposition of the space of all $n \times n$ real symmetric Toeplitz matrices to a finite number of connected components of matrices with a simple spectrum is given. It is then shown that the solvability of the IEPSRTM for all spectra can be deduced if the corresponding map to the IEPSRTM has a nonzero degree for at least one component. This is the case for $n\leqq 4$, which gives an alternative proof to Delsarte and Genin’s results. The IEPSRTM for odd Toeplitz matrices with real-valued entries is also considered.

The Fix-Heiberger procedure for solving the generalized Ill-conditioned symmetric eigenvalue problem

We review a formalism introduced by Fix and Heiberger in 1972 for solving the generalized (or non orthogonal) eigenvalue problem for ill-conditioned symmetric matrices and we discuss its application in quantum chemistry. A few examples dealing with the calculation of high Rydberg orbitals are presented

Algorithm to solve the partial eigenvalue problem for large profile matrices

An algorithm is proposed for computing some of the eigenvalues and the corresponding eigenvectors of the generalized eigenvalue problem for symmetric positive-definite matrices stored in profile form on disk.

Solving the inverse eigenvalue problem via the eigenvector matrix

A numerical algorithm for the inverse eigenvalue problem for symmetric matrices is developed, based on continually updating the eigenvector matrix using plane rotations. A fundamental tool in the algorithm is a matrix formed from the Rayleigh quotients of the eigenvectors with respect to each of the basis matrices involved, with the basis orthonormalized with respect to the Frobenius inner product. Two criteria of closeness to a solution are defined, either of which makes it possible to monitor progress towards a solution. The computational questions involved in this approach are examined in detail. Numerical examples are given for Toeplitz and centrosymmetric tridiagonal matrices. The new algorithm is much more robust than Newton's method.

Solving Additive Inverse Eigenvalue Problems for Symmetric Matrices by the Homotopy Method

Solving Additive Inverse Eigenvalue Problems for Symmetric Matrices by the Homotopy Method