Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

Nonsmooth analysis of eigenvalues

Abstract: The eigenvalues of a symmetric matrix depend on the matrix nonsmoothly. This paper describes the nonsmooth analysis of these eigenvalues. In particular, I present a simple formula for the approximate (limiting Frechet) subdifferential of an arbitrary function of the eigenvalues, subsuming earlier results on convex and Clarke subgradients. As an example I compute the subdifferential of the k'th largest eigenvalue.

A block Lanczos algorithm for computing the q algebraically largest eigenvalues and a corresponding eigenspace of large, sparse, real symmetric matrices

Abstract: Many engineering applications require the computation of the q algebraically largest eigenvalues and a corresponding eigenspace of a large, sparse, real, symmetric matrix. An iterative, block version of the symmetric Lanczos algorithm has been developed for this computation. There are no restrictions on the sparsity pattern within the matrix or on the distribution of the eigenvalues of the matrix. Zero eigenvalues, eigenvalues equal in magnitude but opposite in sign, and multiple eigenvalues can all be handled directly by the procedure.

A Half-Size Singularity Test Matrix for Fast and Reliable Passivity Assessment of Rational Models

Abstract: One major difficulty in the rational modeling of linear systems is that the obtained model can be nonpassive, thereby leading to unstable simulations. The model's passivity properties are usually assessed by computing the eigenvalues of a Hamiltonian matrix, which is derived from the model parameters. The purely imaginary eigenvalues represent crossover frequencies where the model's conductance matrix is singular, allowing to pinpoint frequency intervals of passivity violations. Unfortunately, the eigenvalue computation time can be excessive for large models. Also, the test applies only to symmetrical models, and the testing is made difficult by numerical noise in the extracted eigenvalues. In this paper a new (non-Hamiltonian) half-size singularity test matrix is derived for use with admittance parameter state-space models, which overcomes these shortcomings. It gives a computational speedup by a factor of eight; it is applicable to both symmetric and unsymmetrical models; and it produces noiseless eigenvalues for reliable passivity assessment.

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.