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.

Estimates of eigenvalues of a boundary value problem with a parameter

Abstract: We study an eigenvalue problem for theLaplace operator with boundary condition containing a parameter. We estimate the rate of convergence of the eigenvalues to the eigenvalues of the Dirichlet problem for large positive values of the parameter

On Tridiagonalizing and Diagonalizing Symmetric Matrices with Repeated Eigenvalues

Abstract: We describe a divide-and-conquer tridiagonalization approach for matrices with repeated eigenvalues. Our algorithm hinges on the fact that, under easily constructively verifiable conditions, a symmetric matrix with band width $b$ and $k$ distinct eigenvalues must be block diagonal with diagonal blocks of size at most $b k$. A slight modification of the usual orthogonal band-reduction algorithm allows us to reveal this structure, which then leads to potential parallelism in the form of independent diagonal blocks. Compared to the usual Householder reduction algorithm, the new approach exhibits improved data locality, significantly more scope for parallelism, and the potential to reduce arithmetic complexity by close to 50% for matrices that have only two numerically distinct eigenvalues. The actual improvement depends to a large extent on the number of distinct eigenvalues and a good estimate thereof. However, at worst the algorithms behave like a successive band-reduction approach to tridiagonalization. Moreover, we provide a numerically reliable and effective algorithm for computing the eigenvalue decomposition of a symmetric matrix with two numerically distinct eigenvalues. Such matrices arise, for example, in invariant subspace decomposition approaches to the symmetric eigenvalue problem.

Collaborative spectrum sensing based on upper bound on joint PDF of extreme eigenvalues

Abstract: Detection based on eigenvalues of received signal covariance matrix is currently one of the most effective solution for spectrum sensing problem in cognitive radios. However, the results of these schemes often depend on asymptotic assumptions since the distribution of ratio of extreme eigenvalues is exceptionally mathematically complex to compute in practice. In this paper, a new approach to determine the distribution of ratio of the largest and the smallest eigenvalues is introduced to calculate the decision threshold and sense the spectrum. In this context, we derive a simple and analytically tractable expression for the distribution of the ratio of the largest and the smallest eigenvalues based on upper bound on the joint probability density function (PDF) of the largest and the smallest eigenvalues of the received covariance matrix. The performance analysis of proposed approach is compared with the empirical results. The decision threshold as a function of a given probability of false alarm is calculated to illustrate the effectiveness of the proposed approach.

Transmission eigenvalues for operators with constant coefficients

Abstract: In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.

Bounds for the spectrum of a matrix differential operator with a damping term

Abstract: We consider a second-order matrix ordinary regular differential nonselfadjoint operator with a damping term and selfadjoint boundary conditions. An estimate for the resolvent and bounds for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.