The estimates of diagonally dominant degree and eigenvalues distributions for the Schur complements of matrices
- Yao-tang Li1Email author and
- Feng Wang1
https://doi.org/10.1186/1029-242X-2013-431
© Li and Wang; licensee Springer. 2013
Received: 28 May 2013
Accepted: 26 August 2013
Published: 11 September 2013
Abstract
By applying the properties of Schur complement and some inequality techniques, some new estimates of diagonally dominant degree on the Schur complement of matrices are obtained, which improve the main results of Liu (SIAM J. Matrix Anal. Appl. 27:665-674, 2005) and Liu (Linear Algebra Appl. 432:1090-1104, 2010). As an application, we present some new distribution theorems for eigenvalues of the Schur complement. Finally, we give a numerical example to illustrate the theory results.
MSC:15A45, 15A48.
Keywords
1 Introduction
As in [2], for all and , we call , and the i th dominant degree, α-dominant degree and product α-dominant degree of A, respectively.
is called the Schur complement of A with respect to .
A matrix A is called an M-matrix if there exist a nonnegative matrix B and a real number such that , where is the spectral radius of B. It is well known that A is an H-matrix if and only if is an M-matrix, and if A is an M-matrix, then the Schur complement of A is also an M-matrix and (see [3]). We denote by and the sets of H-matrices and M-matrices, respectively.
The Schur complement has been proved to be a useful tool in many fields such as control theory, statistics and computational mathematics, and many works have been done on it (see [4–8]). Meanwhile, studying the locations of eigenvalues of the Schur complement of matrices is of great significance as shown in [2, 3, 9–14]. In this paper, we present some new estimates of diagonally dominant degree on the Schur complement of matrices and use them to study the distributions for the eigenvalues of the Schur complement of matrices.
The paper is organized as follows. In Section 2, we give several new estimates of the diagonally dominant degree, the α-diagonally dominant degree and product α-diagonally dominant degree on the Schur complement of matrices. In Section 3, several new distribution theorems for eigenvalues of the Schur complements are obtained. In Section 4, we present a numerical example to illustrate the theory results.
2 Estimates of diagonally dominant degree for the Schur complement
In this section, we present several new estimates for the diagonally dominant degree, the α-diagonally dominant degree and product α-diagonally dominant degree of the Schur complement of matrices.
Lemma 1 [5]
If , then .
Lemma 2 [5]
If A is an or A is an , then , i.e., .
Lemma 3 [3]
If A is an or A is an and , then the Schur complement of A is an or an , where is the complement of β in N, and is the cardinality of .
Lemma 4 [2]
Similarly to the proof of Lemma 4 in [13], we can prove that . Thus, we obtain Inequation (1). Similarly, we can prove Inequation (2). □
This shows that Theorem 1 improves Theorem 1 of [13].
Thus, we obtain Inequation (3). Similarly, we can prove Inequation (4). □
This shows that Theorem 2 improves Theorem 2 of [2].
Similarly to the proof of Theorem 2, we can prove the following theorem.
3 Distribution for eigenvalues of the Schur complement
In this section, we present two new distribution theorems for eigenvalues of the Schur complement.
Lemma 5 [2]
Thus, Inequation (5) holds. □
That is, Inequation (6) holds. □
4 A numerical example
In this section, we present a numerical example to illustrate the theory results.
The blue solid line and the green dashed line denote the corresponding discs and , respectively.
The blue solid line and the green dashed line denote the corresponding discs and , respectively.
Declarations
Acknowledgements
The authors are very indebted to the referees for their valuable comments and corrections, which improved the original manuscript of this paper. This work was supported by the National Natural Science Foundation of China (71161020, 11361074) and IRTSTYN.
Authors’ Affiliations
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