On refinement of the generalized Bendixson theorem☆
Introduction
For a given square complex matrix, analyzing algebraic properties and determining geometrical locations for its eigenvalues are not only fundamentally important problems in linear algebra and matrix computations, but also highly valuable requirements in scientific computing and engineering applications. The Geršgorin disk theorem and the Bendixson theorem are the easiest and the most powerful tools in estimating and bounding the eigenvalues of a square matrix [11], [9], [12]. However, these two theorems are not applicable to estimate bounds for the eigenvalues of a matrix pair. To our knowledge, there has been no much discussion on this topic, except for the works of Nakatsukasa [10] and Bai and Ng [8], which present extensions and generalizations for the Geršgorin disk theorem and the Bendixson theorem from a single matrix to a matrix pair under certain restrictions imposed on the considered matrices.
The generalized Bendixson theorem is intimately related to the generalized Rayleigh quotient as well as the matrix-pair generalization of the numerical range. It is particularly useful in analyzing eigen-properties of a preconditioned matrix and the convergence behavior of the corresponding preconditioned Krylov subspace iteration methods for solving large sparse, non-Hermitian, and positive definite systems of linear equations; see, e.g., [1], [2], [3], [4], [5], [6], [7] and the references therein. However, the results given in [8] only tackle the case that the matrices are simultaneously positive (negative) definite of relatively small skew-Hermitian parts, which seriously restricts their application spectrum in numerical analysis and scientific computing. Here we say that a square matrix is positive (negative) definite if its Hermitian part is positive (negative) definite.
In this paper, by in-depth analysis and detailed discussion we further refine and extend the results given in [8], obtaining a complete version of the generalized Bendixson theorem with respect to a square matrix pair. By relaxing the bounds of certain constant quantities involved in the result, we also provide a few simplified versions of this generalized Bendixson theorem, which are more intuitive in description and more convenient in application as well.
The organization of this paper is as follows. After introducing necessary notation, indispensable concepts, and preparatory results in Section 2, we give the refined version of the generalized Bendixson theorem in Section 3. We then present a few simplified and relaxed results in Section 4. Finally, in Section 5, we end the paper with some concluding remarks.
Section snippets
Preliminaries
For a complex number , let and denote its real and imaginary parts, respectively. In particular, if z is a real number, then represents its sign, that is, if z is positive, if z is zero, and otherwise. For a complex matrix , we use to indicate its Hermitian and skew-Hermitian parts, with being the conjugate transpose of the corresponding matrix or vector. Let be the set of all eigenvectors of the
The refinement of the generalized Bendixson theorem
Theorem 2.2 gives lower and upper bounds for the real and imaginary parts of the eigenvalues of the matrix only when the constants and are positive, i.e., the function is positive for all . In addition, Theorem 2.2 only contains the conclusion for the cases , and have the same sign for all , and and have different signs for all . In fact, these assumptions imposed on all vectors are considerably strong and often
The simplified and relaxed results
In this section, by further identifying bounds for the constants ξ, η and ζ, we can simplify the bounds for the real and imaginary parts of the eigenvalues, say, , of the matrix given in both Theorem 3.1, Theorem 3.2.
In fact, when it holds that and when it holds that Therefore, we have
For the constant ξ, if , then In addition, when it holds that and when it
Concluding remarks
Precisely identifying location of the eigenvalues of a square matrix pair is important in theoretical study and valuable in practical applications. The generalized Bendixson theorem established in this work provides a feasible and powerful tool for estimating bounds for the real and imaginary parts of the eigenvalues of a square matrix pair, which also refines and extends the more special and restrictive version given in the earlier work [8] by Bai and Ng.
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This work is dedicated to Professor Hua DAI on the occasion of his 60th birthday. The research of this work is supported by The National Natural Science Foundation (No. 11671393, No. 12071472, and No. 12001043), P.R. China; The Grant of the Government of the Russian Federation (No. 075-15-2019-1928), Russia; and The China-Russia (NSFC-RFBR) International Cooperative Research Project (No. 11911530082 and No. 19-51-53013). The second author is supported by Beijing Institute of Technology Research Fund Program for Young Scholars, P.R. China.