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We consider (and characterize) mainly classes of (positively) stable complex matrices defined via methods of Ger?gorin and Lyapunov. Although the real matrices in most of these classes have already been studied, we sometimes improve upon (and even correct) what has been previously published. Many of the classes turn out quite naturally to be the products of common sets of matrices. A Venn diagram shows how the classes are related. 相似文献
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In this paper, we consider the localization of generalized eigenvalues, and we discuss ways in which the Gersgorin set for generalized eigenvalues can be approximated. Earlier, Stewart proposed an approximation using a chordal metric. We will obtain here an improved approximation, and using the concept of generalized diagonal dominance, we prove that the new approximation has some of the basic properties of the original Ger?gorin set, which makes it a handy tool for generalized eigenvalue localization. In addition, an isolation property is proved for both the generalized Ger?gorin set and its approximation. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
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Vladimir R. Kostić Agnieszka Międlar Ljiljana Cvetković 《Numerical Linear Algebra with Applications》2016,23(2):272-290
The existing algorithms for computing the minimal Ger?gorin set are designed for small and medium size (irreducible) matrices and based on Perron root computations coupled with bisection method and sampling techniques. Here, we first discuss the drawbacks of the existing methods and present a new approach based on the modified Newton's method to find zeros of the parameter dependent left‐most eigenvalue of a Z‐matrix and a special curve tracing procedure. The advantages of the new approach are presented on several test examples that arise in practical applications. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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Fuzhen Zhang 《Linear algebra and its applications》2007,424(1):139-153
This paper aims to set an account of the left eigenvalue problems for real quaternionic (finite) matrices. In particular, we will present the Geršgorin type theorems for the left (and right) eigenvalues of square quaternionic matrices. We shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices. 相似文献
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Geršgorin theorem is a well-known result in eigenvalue localization area. In this paper, using diagonal scaling method, we obtain more Geršgorin-type localizations for the eigenvalues of the Schur complement using the entries of the original matrix instead of the entries of the Schur complement. We deal with classes of matrices with some form of diagonal dominance. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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It is well known that the eigenvalue inclusion domain can be obtained by Geršgorin theorem. It is also known that the Geršgorin theorem is equivalent to the statement that each strictly diagonally dominant (SDD) matrix is nonsingular. Similarly, statements about nonsingularity of some classes of matrices, which are generalizations of SDD class, produce new Geršgorin‐type theorems for eigenvalue inclusion domain. In this paper we will consider some subclasses of H‐matrices and corresponding (Geršgorin‐type) eigenvalue inclusion theorems. Finally, we will establish relationship between given inclusion domains. 相似文献
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Givens' transformation (1954) was originally applied to real matrices. We shall give an extension to quaternion valued matrices. The complex case will be treated in the introduction. We observe that the classical Givens' rotation in the real and in the complex case is itself a quaternion using an isomorphism between certain (2×2) matrices and R
4 equipped with the quaternion multiplication. In the real and complex case Givens' (2×2) matrix is determined uniquely up to an arbitrary (real or complex) factor with ||=1. However, because of the noncommutativity of quaternions, we shall show that in the quaternion case such a factor must obey certain additional restrictions. There are two numerical examples including a MATLAB program and some hints for implementation. Matrices with quaternion entries arise, e.g., in quantum mechanics.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
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This paper aims to discuss the location for right eigenvalues of quaternion matrices. We will present some different Gerschgorin type theorems for right eigenvalues of quaternion matrices, based on the Gerschgorin type theorem for right eigenvalues of quaternion matrices (Zhang in Linear Algebra Appl. 424:139?C153, 2007), which are used to locate the right eigenvalues of quaternion matrices. We shall conclude this paper with some easily computed regions which are guaranteed to include the right eigenvalues of quaternion matrices in 4D spaces. 相似文献
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Ljiljana Cvetkovi? Vladimir Kosti? Rafael Bru Francisco Pedroche 《Advances in Computational Mathematics》2011,35(2-4):271-280
It is well known that the spectrum of a given matrix A belongs to the Ger?gorin set ??(A), as well as to the Ger?gorin set applied to the transpose of A, ??(A T ). So, the spectrum belongs to their intersection. But, if we first intersect i-th Ger?gorin disk ?? i (A) with the corresponding disk $\Gamma_i(A^T)$ , and then we make union of such intersections, which are, in fact, the smaller disks of each pair, what we get is not an eigenvalue localization area. The question is what should be added in order to catch all the eigenvalues, while, of course, staying within the set ??(A)??????(A T ). The answer lies in the appropriate characterization of some subclasses of nonsingular H-matrices. In this paper we give two such characterizations, and then we use them to prove localization areas that answer this question. 相似文献
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Motivated by conditions that arise from results on mean first passage times matrices in Markov chains, we consider here two classes of real matrices whose elements satisfy some of these conditions, or variation thereof, and which result in the nonsingularity of their elements. The conditions are quite distinct from Ger?gorin circles-type conditions. Our results lead to a sufficient condition for matrices to have 1 as their unique positive eigenvalue. 相似文献
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Istkhar Ali 《Journal of Applied Mathematics and Computing》2018,58(1-2):323-334
In this paper, localization theorems for left and right eigenvalues of a quaternion matrix are presented. Some differences between quaternion matrices and split quaternion matrices are summarized. A counter example for Gerschgorin theorems for left and right eigenvalues of a split quaternion matrix is given. Finally, a method for finding right eigenvalues of a split quaternion matrix pencil is presented. 相似文献
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P. Catarino 《Journal of Difference Equations and Applications》2016,22(2):343-351
In this paper we consider a g – circulant, right circulant, left circulant and a special kind of a tridiagonal matrices whose entries are h(x) – Fibonacci quaternion polynomials. We present the determinant of these matrices and with the tridiagonal matrices we show that the determinant is equal to the nth term of the h(x) – Fibonacci quaternion polynomial sequences. 相似文献
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Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m ( n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages. 相似文献
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Xue Ding 《Linear and Multilinear Algebra》2013,61(9):1737-1749
In this paper, we study the spectral properties of the large self-dual dilute quaternion random matrices. For the dilute case, we prove that the empirical spectral distribution still converges to the semicircular law with some appropriate normalization. Further, we obtain the limits of the extreme eigenvalues of the large self-dual dilute quaternion random matrices under some moment assumptions of the underlying distributions and give a necessary condition for the strong convergence of the extreme eigenvalues. 相似文献
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本文给出了四元数矩阵惯性的定义,讨论了四元数体上Lyapunov矩阵方程的唯一解,推广了一般惯性定理、Lyapunov稳定性定理、Carlson-Schneider定理、Stein稳定性定理等一些重要的结果到四元数矩阵,同时得出了四元数体上稳定矩阵的一些判别条件. 相似文献
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A proof of the Ger?gorin disk theorem is given, in which the only analytic tool used is a contour integral. 相似文献