Showing papers on "Spectrum of a matrix published in 1988"

The eigenvalues of second-order spectral differentiation matrices*

Abstract: The eigenvalues of the pseudospectral second derivative matrix with homogeneous Dirichlet boundary conditions are important in many applications of spectral methods. This paper investigates some of their properties. Numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately, but the errors in the remaining ones are large. It is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid. In particular, it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a Chebyshev distribution of nodes, and six points per wavelength for a uniform distribution. These results are in agreement with the observed fractions of accurate eigenvalues. By using the characteristic polynomial, a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2% or better. The effect of filtering on the eigenvalues is studied numerically.

Eigenvalues of the eight-vertex model transfer matrix and the spectrum of theXYZ Hamiltonian

Abstract: We study the eigenvalue problem for the transfer matrix of the eight-vertex model. By using an inversion relation which was recently discovered we develop a new method to calculate all eigenvalues of the transfer matrix in the thermodynamic limit. This leads to a complete classification of the spectrum. The results are used to determine all energy excitations of theXYZ-model.

Eigenvalues of the moment-method matrix and their effect on the convergence of the conjugate gradient algorithm (EM scattering)

Abstract: A theory that relates eigenvalues of a continuous operator to those of the moment-method matrix operator is discussed and confirmed by examples. This theory suggests reasons for ill conditioning when certain types of basis and testing functions are used. In addition, the effect of eigenvalue location on the convergence of the conjugate gradient (CG) method is studied. The convergence rate of the CG method is dependent on the eigenvalues of the iteration matrix as well as on the number of eigenvectors of the iteration matrix needed to represent the right side of the equation. These findings explain the previously reported convergence behavior of the CG method when applied to electromagnetic-scattering problems. >

Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems

Abstract: In this paper, a sufficient and necessary condition for the asymptotic stability of the discrete linear interval system X(k + 1) = A1X(k) is presented; i.e. all the eigenvalues of each matrix AϵA1, (an interval matrix) have magnitudes less than 1 if and only if all the eigenvalues of every ‘extreme matrix’ of the interval matrix A1, have magnitudes less than 1.

An improved Toeplitz approximation method

Abstract: The authors suggest a modification of the Toeplitz approximation method for estimating frequencies of multiple sinusoids from covariance measurements. The method constructs a state-feedback matrix following a low-rank approximation of the Toeplitz covariance matrix via singular-value decomposition. Ideally, the eigenvalues of this state-feedback matrix will be on the unit circle in the complex plane, and the angles that they make with the real axis will be equal to the unknown sinusoid frequencies. The modification proposed here exploits this prior knowledge of the modulus of the eigenvalues and guarantees that even in the presence of noise the eigenvalues of the estimated state-feedback matrix will lie on the unit circle. >

Equivalence of transfer matrices

Abstract: Three theorems dealing with transfer matrices in statistical mechanical systems are proved. The theorems state that the nonzero eigenvalues of transfer matrices formed through various prescriptions are identical. Hence it is possible to ascribe a physical meaning to all the eigenvalues of a transfer matrix, not just to the few largest eigenvalues. The first theorem states that the transfer matrix formed by building a system M layers at a time has as its only nonzero eigenvalues the eigenvalues of the transfer matrix formed by building the M layers of the system one at a time. This theorem relates the product of two nM×nM M‐layer transfer matrices to the product of M one‐layer M×M transfer matrices. The second theorem states that one of the nM×nM M‐layer transfer matrices (for M>1) has only one nonzero eigenvalue. A procedure for finding this eigenvalue and all eigenvectors is given. The third theorem generalizes the first to the case where the chosen layering is not an integer multiple of the interaction ...

Coupled transmission lines revisited

Abstract: The author presents a very satisfying and disarmingly simple way of introducing students to the parallel line coupler, which derives its S matrix directly from a knowledge of its eigenvectors and eigenvalues. For a pair of symmetrical coupled lines, the eigenvectors can be found by inspection, and the eigenvalues determined easily by means of flow graphs. The elements of the S matrix are then known, being linear combinations of the eigenvalues. >

The eigenvalues of second-order spectral

Abstract: J. A. C. WEIDEMANt AND L. N. TREFETHENt Abstract. The eigenvalues of the pseudospectral second derivative matrix with homogeneous Dirichlet boundary conditions are important in many applications of spectral methods. This paper investigates some of their properties. Numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately, but the errors in the remaining ones are large. It is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid. In particular, it is proved that r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a Chebyshev distribution of nodes, and six points per wavelength for a uniform distribution. These results are in agreement with the observed fractions of accurate eigenvalues. By using the characteristic polynomial, a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2% or better. The effect of filtering on the eigenvalues is studied numerically.

Bounds on decay constants for diffusion through inhomogeneous media

Abstract: The decay constants for diffusion through inhomogeneous media are known to be proportional to the eigenvalues of the corresponding elliptic operator. A new method of obtaining a hierarchy of upper bounds on sums and products of these eigenvalues as well as the eigenvalues themselves is presented. The first member of this hierarchy is just the usual Rayleigh-Ritz quotient. The other members of the hierarchy are generalised Rayleigh-Ritz quotients which can be derived simply using properties of integrals of the solutions of the diffusion equation. Explicit bounds are presented for the first three eigenvalues, but general methods of obtaining bounds for higher-order eigenvalues are also outlined. For fixed time t, many of the bounds reduce to results given by the classical method of moments. The hierarchy of rigorous variational bounds on the eigenvalues studied may be generated using simple recursion relations based on properties of the characteristic orthogonal polynomials. The conditions on the trial functions used to obtain bounds on eigenvalues higher than the first are much simpler than those required by the traditional Rayleigh-Ritz procedure.

Number of positive eigenvalues of the solution to the discrete Lyapunov matrix equation

Abstract: It is known that the discrete Lyapunov matrix equation has a positive semidefinite symmetric solution under an appropriate condition. This note proves the fact that the number of positive eigenvalues of the solution is equal to the rank of the controllability matrix formed from the coefficient matrices.