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.

On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator

Abstract: On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.

A theorem on the Lyapunov matrix equation

Abstract: Given the Lyapunov matrix equation A'P + PA + 2\sigmaQ = 0 where σ is some positive scalar, a necessary and sufficient condition for the real parts of the eigenvalues of A to be less than -σ is that P - Q is negative definite. The condition provides an upper bound to the solution of the Lyapunov matrix equation and is useful in the design of minimum-time or minimum-cost linear control systems.

A dynamical system that computes eigenvalues and diagonalizes matrices with a real spectrum

Abstract: The present paper deals with the problem of diagonalizing matrices using a control system of the form A = [U, A], where [U, A] = UA - AU and A, U are real matrices. It is shown that the feedback U = [N, A + AT] + p[AT, A], N diagonal, rho > 0 allows to solve the diagonalization problem under the assumption that the to be diagonalized matrix has real spectrum. Moreover, in the case of a complex spectrum, the feedback allows to check if a matrix is stable or to compute all eigenvalues of a matrix or roots of a polynomial.

On Inclusion and Exclusion Intervals for the Real Eigenvalues of Real Matrices

Abstract: A real square matrix with positive row sums and all its off-diagonal elements bounded below by the corresponding row means is called a $C$-matrix, which is introduced by Pena [Exclusion and inclusion intervals for the real eigenvalues of positive matrices, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 908-917]. In this paper, a new class of nonsingular matrices—$MC$-matrices containing $C$-matrices—is first defined. By properties of a subclass of $MC$-matrices, we present new exclusion intervals of the real eigenvalues of a real matrix, which are further applied to localize the real eigenvalues different from 1 of a positive stochastic matrix. Secondly, an inclusion interval for the real parts of eigenvalues of a real matrix is established. Finally, for real matrices with nonnegative off-diagonal elements, lower and upper bounds of real eigenvalues are obtained. Furthermore, sufficient conditions are derived to indicate that the real inclusion intervals provided by Pena [On an alternative to Gerschgorin circles and ovals of Cassini, Numer. Math., 95 (2003), pp. 337-345] are subsets of those provided by the ovals of Cassini.