Spectrum Localization for Almost Skew-symmetric Matrices

Linear Algebra and its Applications, 1992

A complex matrix is said to be stable if all its eigenvalues have negative real part. Let J be a Jordan block with zeros on the diagonal and ones on the superdiagonal, and consider analytic matrix perturbations of the form A() = J + B + O(2), where is real and positive. A necessary condition on B for the stability of A() on an interval (0; 0), and a su cient condition on B for the existence of such a family A(), is (i) Re tr B 0; (ii) the sum of the elements on the rst subdiagonal of B has nonpositive real part and zero imaginary part; (iii) the sum of the elements on each of the other subdiagonals of B is zero. This result is extended to matrices with any number of nonderogatory eigenvalues on the imaginary axis, and to a stability de nition based on the spectral radius. A generalized necessary condition, though not a su cient condition, applies to arbitrary Jordan structures. The proof of our results uses two important techniques: the Puiseux-Newton diagram and the normal form of Arnold. In the nonderogatory case our main results were obtained by Levantovskii in 1980 using a di erent proof. Practical implications are discussed.

Linear Algebra and its Applications, 2006

The analysis of the Perron eigenspace of a nonnegative matrix A whose symmetric part has rank one is continued. Improved bounds for the Perron root of Levinger's transformation (1 − α)A + αA t (α ∈ [0, 1]) and its derivative are obtained. The relative geometry of the corresponding left and right Perron vectors is examined. The results are applied to tournament matrices to obtain a comparison result for their spectral radii.

Linear Algebra and its Applications, 1999

In [18], among other equivalent conditions, it is proved that a square complex matrix A is permutationally similar to a block-shift matrix if and only if for any complex matrix B with the same zero pattern as A, W (B), the numerical range of B, is a circular disk centered at the origin. In this paper, we add a long list of further new equivalent conditions. The corresponding result for the numerical range of a square complex matrix to be invariant under a rotation about the origin through an angle of 2π/m, where m 2 is a given positive integer, is also proved. Many interesting by-products are obtained. In particular, on the numerical range of a square nonnegative matrix A, the following unexpected results are established: (i) when the undirected graph of A is connected, if W (A) is a circular disk centered at the origin, then so is W (B), for any complex matrix B with the same zero pattern as A; (ii) when A is irreducible, if λ is an eigenvalue in the peripheral spectrum of A that lies on the boundary of W (A), then λ is a sharp point of W (A). We also obtain results on the numerical range of an irreducible square nonnegative matrix, which strengthen or clarify the work of Issos [9] and Nylen and Tam [14] on this topic. Open questions are posed at the end.

Axioms, 2021

The paper is a survey of the recent results of the author on the perturbations of matrices. A part of the results presented in the paper is new. In particular, we suggest a bound for the difference of the determinants of two matrices which refines the well-known Bhatia inequality. We also derive new estimates for the spectral variation of a perturbed matrix with respect to a given one, as well as estimates for the Hausdorff and matching distances between the spectra of two matrices. These estimates are formulated in the terms of the entries of matrices and via so called departure from normality. In appropriate situations they improve the well-known results. We also suggest a bound for the angular sectors containing the spectra of matrices. In addition, we suggest a new bound for the similarity condition numbers of diagonalizable matrices. The paper also contains a generalization of the famous Kahan inequality on perturbations of Hermitian matrices by non-normal matrices. Finally, ta...

Mathematics of Computation, 1990

Let A be a nonsingular matrix with positive inverse and B a non-negative matrix. Let the inverse of A + v B A + vB be positive for 0 ≤ v > v ∗ > + ∞ 0 \leq v > {v^ \ast } > + \infty and at least one of its entries be equal to zero for v = v ∗ v = {v^ \ast } ; an algorithm to compute v ∗ {v^ \ast } is described in this paper. Furthermore, it is shown that if A + A T A + {A^{\text {T}}} is positive definite, then the inverse of A + v ( B − B T ) A + v(B - {B^{\text {T}}}) is positive for 0 ≤ v > v ∗ 0 \leq v > {v^ \ast } .

Random Matrices: Theory and Applications, 2016

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index are interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.

A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J--sign-symmetric. The spectrum of a J--sign-symmetric irreducible matrix is studied under assumptions that its second compound matrix is also J--sign-symmetric and irreducible. The conditions, when such matrices have complex eigenvalues on the largest spectral circle, are given. The existence of two positive simple eigenvalues $\lambda_1 > \lambda_2 > 0$ of a J--sign-symmetric irreducible matrix A is proved under some additional conditions. The question, when the approximation of a J--sign-symmetric matrix with a J--sign-symmetric second compound matrix by strictly J--sign-symmetric matrices with strictly J--sign-symmetric compound matrices is possible, is also studied in this paper. Comment: 24 pages

Linear and Multilinear Algebra, 2009

In the literature it is known that the decomposable numerical range We construct a symmetric unitary matrix A ∈ C n×n such that the decomposable numerical range W ∧ k (A) is not star-shaped and hence not simply connected. We then consider a real analog R ∧ k (A) and show that R ∧ k (A) is star-shaped if A ∈ C n×n is skew symmetric. Such star-shapedness result is also true for the Pfaffian numerical range P ∧ k (A).

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENTATIONES MATHEMATICAE UNIVERSITAT1S CAROLINAE 21,2 (1980) A NOTE ON ESTIMATE OF THE SPECTRAL RADIUS OF SYMMETRIC MATRICES

Using the notions of the numerical range, Schur complement and unitary equivalence, an eigenvalue inequality is obtained for a general complex matrix, giving rise to a region in the complex plane that contains its spectrum. This region is determined by a curve, generalizing and improving classical eigenvalue bounds obtained by the Hermitian and skew-Hermitian parts, as well as the numerical range of a matrix.