The improved disc theorems for the Schur complements of diagonally dominant matrices
© Zhang et al.; licensee Springer 2013
Received: 14 October 2011
Accepted: 14 December 2012
Published: 3 January 2013
The theory of Schur complement is very important in many fields such as control theory and computational mathematics. In this paper, applying the properties of Schur complement, utilizing some inequality techniques, some new estimates of diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. Further, as an application of these derived results, we present some distributions for the eigenvalues of the Schur complements. Finally, the numerical example is given to show the advantages of our derived results.
The Schur complement has been proved to be a useful tool in many fields such as numerical algebra and control theory. [1, 2] proposed a kind of iteration called the Schur-based iteration. Applying this method, we can solve large scale linear systems though reducing the order by the Schur complement. In addition, when utilizing the conjugate gradient method to solve large scale linear systems, if the eigenvalues of the system matrix are more concentrated, the convergent speed of the iterative method is faster (see, e.g., [, pp.312-317]). From [1, 2], it can be seen that for large scale linear systems, after applying the Schur-based iteration to reduce the order, the corresponding system matrix of linear equations is the Schur complement of the system matrix of original large scale linear systems and its eigenvalues are more concentrated than those of the original system matrix, leading to the Schur-based conjugate gradient method computing faster than the ordinary conjugate gradient method.
Hence, it is always interesting to know whether some important properties of matrices are inherited by their Schur complements. Clearly, the Schur complements of positive semidefinite matrices are positive semidefinite, the same is true for M-matrices, H-matrices and inverse M-matrices (see [4, 5]). Carlson and Markham showed that the Schur complements of strictly diagonally dominant matrices are diagonally dominant (see ). Li, Tsatsomeros and Ikramov independently proved the Schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant (see [7, 8]). These properties have been repeatedly used for the convergence of iterations in numerical analysis and for deriving matrix inequalities in matrix analysis (see [3, 9, 10]). More importantly, the distribution for the eigenvalues of the Schur complement is of great significance, as shown in [1, 2, 8, 11–17]. The aim of this paper is to study the distributions for the eigenvalues of the Schur complement of some diagonally dominant matrices.
If all inequalities in (1.1)-(1.4) are strict, then A is said to be a strictly (row) diagonally dominant matrix (), a strictly doubly diagonally dominant matrix (), a strictly γ-diagonally dominant matrix () and a strictly product γ-diagonally dominant matrix (), respectively.
A matrix A is an M-matrix if it can be written in the form of with P being nonnegative and , where denotes the spectral radius of P. A matrix A is a H-matrix if is a M-matrix. We denote by and the sets of H- and M-matrices, respectively.
is called the Schur complement of A with respect to .
The paper is organized as follows. In Section 2, we give several new estimates of diagonally dominant degree on the Schur complement of matrices, which improve some relative results. In Section 3, as an application of these derived results, the distributions for eigenvalues are obtained. In Section 4, we give a numerical example to show the advantages of our derived results.
2 The diagonally dominant degree for the Schur complement
In this section, we give several new estimates of diagonally dominant degree on the Schur complement of matrices, which improve some relative results.
Lemma 1 
If A is an H-matrix, then .
Lemma 2 
If or , then , i.e., .
Lemma 3 
If or and , then the Schur complement of A is in or , where is the Schur complement of α in N and is the cardinality of .
Lemma 4 
Let , thus we easily get (2.1). Similarly, we obtain (2.2). □
This means that Theorem 1 improves Theorem 1 of .
Take in (2.5). By (2.6), thus it is not difficult to get that (2.7) follows. Similarly, we obtain (2.8). □
It is known that the Schur complements of diagonally dominant matrices are diagonally dominant (see [12, 13]). However, this property is not always true for γ-diagonally dominant matrices and for product γ-diagonally dominant matrices, as shown in .
In the sequel, we obtain some disc separations for the γ-diagonally and product γ-diagonally dominant degree of the Schur complement, from which we provide that the Schur complement of the γ-diagonally and product γ-diagonally dominant matrices is also γ-diagonally dominant and product γ-diagonally under some restrictive conditions.
Thus we get (2.9). Similarly, we have (2.10). □
This means that Theorem 3 improves Theorem 2 of .
In a similar way to the proof of Theorem 3, we get the following theorem immediately.
3 Distribution for eigenvalues
In this section, as an application of our results in Section 2, we present some locations for the eigenvalues of the Schur complements.
Remark 3 By Remark 2, it is obvious that Theorem 5 improves Theorem 3 of .
In a similar way to the proof of Theorem 5, we obtain the following theorem according to Theorem 2.
Next, we obtain some distributions for the eigenvalues of the Schur complements of matrices under the conditions such as degree.
Lemma 5 
Thus (3.4) holds. □
Remark 4 By Remark 2, it is obvious that Theorem 7 improves Theorem 4 of .
4 A numerical example
In this section, we show how to estimate the bounds for eigenvalues of the Schur complement with the elements of the original matrix to show the advantages of our results.
It is clear that from both (4.1), (4.2) and Figure 1.
It is clear that from both (4.3), (4.4) and Figure 2.
The work was supported in part by the National Natural Science Foundation of China (10971176), the Program for Changjiang Scholars and Innovative Research Team in University of China (No. IRT1179), the Key Project of Hunan Provincial Natural Science Foundation of China (10JJ2002), the Key Project of Hunan Provincial Education Department of China (12A137), the Hunan Provincial Innovation Foundation for Postgraduate (CX2011B242) and the Aid Program Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province of China.
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