Coincident Eigenvalues Research Articles

The Structure and Distribution of Roots of 2×2 Matrices

This paper is concerned with Jordan canonical form theorem of algebraic formulae giving all the solutions of the matrix equation Xm= A where n is a positive integer greater than 2 and A is a 2 × 2 matrix with real or complex elements. If A is a 2 × 2 non-singular matrix, the equation Xm = A has infinitely many solutions and we obtain explicit formulae giving all the solutions. If A is a 2 × 2 singular matrix, and we obtained necessary and sufficient condition of square root . This leads to very simple formulae for all the solutions when A is either a singular matrix or a non-singular matrix with two coincident eigenvalues. We also determine the precise number of solutions in various cases.

Extraction of nth roots of 2×2 matrices

This paper is concerned with the determination of algebraic formulae giving all the solutions of the matrix equation X n = A where n is a positive integer greater than 2 and A is a 2×2 matrix with real or complex elements. If A is a 2×2 scalar matrix, the equation X n = A has infinitely many solutions and we obtain explicit formulae giving all the solutions. If A is a non-scalar 2×2 matrix, the equation X n = A has a finite number of solutions and we give a formula expressing all solutions in terms of A and the roots of a suitably defined nth degree polynomial in a single variable. This leads to very simple formulae for all the solutions when A is either a singular matrix or a non-singular matrix with two coincident eigenvalues. Similarly when n=3 or 4, we get explicit algebraic formulae for all the solutions. We also determine the precise number of solutions in various cases.

Survey and Review

It is a great pleasure to assume the role of Section Editor for the Survey and Review section of this journal. I'd like to thank Nick Trefethen for editing this section so ably over the past four years, and guiding its development since the format of SIAM Review was substantially overhauled in 1999. I will have the help of an excellent editorial board, and we intend to continue the tradition of publishing informative and well-written papers on topics of broad interest to the applied mathematics community. I encourage prospective authors to read the revised guidelines at the front of this journal. SIAM Review has played a preeminent role in the Society for Industrial and Applied Mathematics since Volume 1 was published in 1959. Over the years many important papers have appeared in these pages, and as part of SIAM's 50th anniversary celebration we thought it would be appropriate to reprint a classic SIAM Review paper from the past. The one we have chosen, "Nineteen Dubious Ways to Compute the Exponential of a Matrix," by Cleve Moler and Charles Van Loan, is one of the most frequently cited papers ever to appear in this journal. More importantly, it's still worth reading, and worth introducing to a new generation of students and young researchers. It's a great paper for a journal club; working through the nineteen dubious ways gives a tour of a broad spectrum of techniques and sets the stage for discussing and exploring many related issues. Pade approximation, ODE solvers, the Cayley--Hamilton theorem, polynomial interpolation, Laplace transforms, the Jordan canonical form, Schur decompositions, and the Trotter product formula---these are just some of the approaches that make an appearance. Understanding why they are dubious leads to the study of conditioning and stability, inverse error analysis, logarithmic norms, and an appreciation for the effects of nonnormal matrices and coincident eigenvalues. All this in 37 very readable pages. Of course, much progress has been made in the years since this paper first appeared. Many of the issues discussed are better understood today than they were at the time, and new algorithms have appeared that may be less dubious. The behavior of nonnormal matrices and the "hump problem" are clarified via the theory of pseudospectra and related notions. Particularly noteworthy on the algorithmic front is the development of Krylov space methods that give good accuracy and efficient computation of the exponential of large matrices. These methods are now frequently used in various contexts, such as quantum dynamics, Markov chains, and differential equations. The original "Nineteen Ways" paper is an excellent starting point for exploring a host of other topics. To aid in this exploration, Moler and Van Loan have kindly provided a summary of some recent developments, with a few entry points into the vast literature that has appeared since the original paper. Their new contribution follows the original paper, starting on page 40.

Optical beams with complex astigmatism in the case of coincident eigenvalues of the cavity ray matrix

An analysis is carried out of the optical cavities whose natural modes have the form of Gaussian beams with complex astigmatism. In the case of coincident eigenvalues of the cavity ray matrix, the wave beams were found to be determined not only by the geometrical optics characteristics of a cavity, but also by additional cavity-independent parameters. A detailed analysis is given for the fundamental mode of such cavities. Concrete cavities possessing these properties are noted.

Effective Mass Sensitivities for Systems with Repeated Eigenvalues

A generalization for the calculation of effective mass sensitivities to the case of coincident eigenvalues has been proposed. The results obtained with the exact approach and an approach based on a suitable perturbation, introduced to eliminate the eigenvalue multiplicity, are in excellent agreement. The two approaches based on the finite dfference approximation are still in good agreement, but as expected, the elimination of significant digits does affect the precision

Eigensolutions of perturbed nearly defective matrices

Abstract Perturbation methods for approximating eigenvalues and eigenvectors of perturbed nearly defective (mistuned) systems have been discussed. It is shown that the mistuning causes Taylor expansions to be not uniformly valid even in small intervals of the perturbation parameter, subsequently rendering them useless for practical purposes. The problem is overcome by starting the perturbation expansion from an exactly defective (tuned) system “close” to the mistuned one. Asymptotic expansions are then obtained in terms of fractional powers of two parameters: the modification and the mistuning parameter. However, as the tuned system is unknown, an inverse problem has to be solved in order to determine it. An algorithm valid for the particular but frequent case of several couples of nearly coincident eigenvalues has been detailed for this problem; an outline of a more general case is also given. Illustrative examples are presented.

More efficient use of determinants to solve transcendental structural eigenvalue problems reliably

The authors have previously presented the ‘multiple determinant parabolic interpolation method’ for solving the transcendental eigenvalue problems which arise when the ‘exact’ member equations obtained by solving the governing differential equations of members are used to find the eigenvalues of structures. For non-coincident eigenvalues this method achieves substantial time savings over bisection and is faster than other competing methods. The present paper extends the method to groups of coincident eigenvalues, for which it achieves comparable time savings while retaining the efficiency of the original method for other problems including those with close eigenvalues.

Optimal structural design under multiple eigenvalue constraints

With increasing complexity, structures that are optimally designed subject to an eigenvalue constraint (buckling load, fundamental frequency, etc.) are likely to display multiple coincident eigenvalues. For example, optimal statically determinate and simply indeterminate columns buckle into a single mode, whereas fixed-fixed columns may exhibit a dual buckling mode. This phenomenon has been observed first by Olhoff and Rasmussen[1].Even if the optimal design is in the interior of the admissible design space, multiple eigenvalue solutions are not stationary but singular. The nature of the singularity is the main topic of this investigation. Necessary and sufficient conditions for local and global optimality are explored and explicit optimality criteria are established for a double mode solution. In that case the space of all admissible design changes is split into a two-dimensional subspace in which the dual eigenvalues separate, and a complementary subspace in which they remain coincident. It is within this latter subspace that sequential approximations must take place.The criteria developed here are applied first to a two-degree-of freedom system. In addition, an exact analytical solution is established for the fixed-fixed column problem. The accuracy of the numerical solution of this problem in [1] had been challenged but is now confirmed.The paper ends with a discussion of the ring buckling problem. The prismatic design is shown to be optimal without being stationary; however, unlike the cases discussed previously the optimal eigenvalue for the ring corresponds to a continuous spectrum of buckling modes.

Mode Finding in Nonlinear Structural Eigenvalue Calculations

Abstract The eigenvalue problems resulting from stiffness matrix formulations of structural vibration and buckling problems are nonlinear if substructures are analyzed exactly, or if classical frequency (vibration problems) or load factor (buckling problems) dependent member equations are used. This makes rapid calculation of accurate free vibration or buckling modes difficult. This paper presents several techniques which might overcome this difficulty, examines them theoretically and experimentally, and gives some of the ways in which the more successful techniques can be incorporated in mode finding methods. Coincident eigenvalues (i.e., natural frequencies or critical load factors) are included.