Eigenvalues Of Complex Matrices Research Articles

On sets of eigenvalues of matrices with prescribed row sums and prescribed graph

Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore, we completely characterize the sets of possible Perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study. Extending known results to the reducible case, we derive new sharp bounds on the set of eigenvalues or Perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries. In the last section of the paper a new constructive proof of the Camion–Hoffman theorem is given.

Strict double diagonal dominance in Euclidean Jordan algebras

In the first part of the paper, we deal with Euclidean Jordan algebraic generalizations of some results of Brualdi on inclusion regions for the eigenvalues of complex matrices using directed graphs. As a consequence, the theorems of Brauer–Ostrowski and Brauer on the location of eigenvalues are extended to the setting of Euclidean Jordan algebras. In the second part, motivated by the work of Li and Tsatsomeros on the class of doubly diagonally dominant matrices with complex entries and its subclasses, we present some inter-relations between the H-property, generalized strict diagonal dominance, invertibility, and strict double diagonal dominance in Euclidean Jordan algebras. In addition, we show that in a Euclidean Jordan algebra, the Schur complements of a strictly doubly diagonally dominant element inherit this property.

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Geršgorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2 × 2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Geršgorin.

Ergodicity Coefficients Defined by Vector Norms

Ergodicity coefficients for stochastic matrices determine inclusion regions for subdominant eigenvalues; estimate the sensitivity of the stationary distribution to changes in the matrix; and bound the convergence rate of methods for computing the stationary distribution. We survey results for ergodicity coefficients that are defined by p-norms, for stochastic matrices as well as for general real or complex matrices. We express ergodicity coefficients in the one-, two-, and infinity-norms as norms of projected matrices, and we bound coefficients in any p-norm by norms of deflated matrices. We show that two-norm ergodicity coefficients of a matrix A are closely related to the singular values of A. In particular, the singular values determine the extreme values of the coefficients. We show that ergodicity coefficients can determine inclusion regions for subdominant eigenvalues of complex matrices, and that the tightness of these regions depends on the departure of the matrix from normality. In the special case of normal matrices, two-norm ergodicity coefficients turn out to be Lehmann bounds.

Convexity and Lipschitz Behavior of Small Pseudospectra

The e-pseudospectrum of a matrix $A$ is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance e of $A$, measured by the operator 2-norm. Given a nonderogatory matrix $A_0$, for small $\epsilon > 0, we show that the e-pseudospectrum of any matrix $A$ near $A_0$ consists of compact convex neighborhoods of the eigenvalues of $A_0$. Furthermore, the dependence of each of these neighborhoods on $A$ is Lipschitz.

Coupling of eigenvalues of complex matrices at diabolic and exceptional points

The paper presents a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two physical examples illustrate effectiveness and accuracy of the presented theory.

Robust stability and a criss-cross algorithm for pseudospectra

x = Ax is robustly stable when all eigenvalues of complex matrices within a given distance of the square matrix A lie in the left half-plane. The ‘pseudospectral abscissa’, which is the largest real part of such an eigenvalue, measures the robust stability of A .W epresent an algorithm for computing the pseudospectral abscissa, prove global and local quadratic convergence, and discuss numerical implementation. As with analogous methods for calculating H∞ norms, our algorithm depends on computing the eigenvalues of associated Hamiltonian matrices.

A note on unifying absolute and relative perturbation bounds

Perturbation bounds for invariant subspaces and eigenvalues of complex matrices are presented that lead to absolute as well as a large class of relative bounds. In particular it is shown that absolute bounds (such as those by Davis and Kahan, Bauer and Fike, and Hoffman and Wielandt) and some relative bounds are special cases of `universal' bounds. As a consequence, we obtain a new relative bound for subspaces of normal matrices, which contains a deviation of the matrix from (positive-) definiteness. We also investigate how row scaling affects eigenvalues and their sensitivity to perturbations, and we illustrate how the departure from normality can affect the condition number (with respect to inversion) of the scaled eigenvectors.

A GENERALIZATION OF GERSHGORIN'S THEOREM

Estimates convenient for applicatins are obtained for the eigenvalues of complex matrices. Some applications in stability theory are indicated. The results are obtained by methods of convex analysis and the theory of nonnegative matrices. Figures: 2. Bibliography: 24 titles.

Computing Eigenvalues of Complex Matrices by Determinant Evaluation and by Methods of Danilewski and Wielandt

Computing Eigenvalues of Complex Matrices by Determinant Evaluation and by Methods of Danilewski and Wielandt