The Estimates of Diagonally Dominant Degree and Eigenvalue Inclusion Regions for the Schur Complement of Matrices

The theory of Schur complement plays an important role in many fields such as matrix theory, control theory and computational mathematics. In this paper, some new estimates of diagonally, α-diagonally and product α-diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. As an application, we present several new eigenvalue inclusion regions for the Schur complement of matrices. Finally, we give a numerical example to illustrate the advantages of our derived results.


Introduction
. We write ( ) ( ) , , , We know that A is called a strictly diagonally dominant matrix if ( ) > , .
ii i a

R i N ∀ ∈ A
A is called an Ostrowski matrix (see [1]) if and n OS will be used to denote the sets of all n n × strictly diagonally dominant matrices and the sets all n n × Ostrowski matrices, respectively.As shown in [2], for 1 i n ≤ ≤ and ( ) ( ) ( ) ( ) ( ) A the i-th diagonally, α-diagonally and product α-diagonally dominant degree of A, respectively.
For N β ⊆ , denote by β the cardinality of β and

A
is the submatrix of A with row indices in β and column indices in γ .In particular, ( ) is called the Schur complement of A with respect to ( ) The comparison matrix of A, ( ) ( ) µ A is an m-matrix, and if A is an m-matrix, then the Schur complement of A is also an m-matrix and det 0 > A (see [3]).We denote by H n and M n the sets of h-matrices and m-matrices, respectively.The Schur complement of matrix is an important part of matrix theory, which has been proved to be useful tools in many fields such as control theory, statistics and computational mathematics.A lot of work has been done on it (see [4]- [8]).We know that the Schur complements of strictly diagonally dominant matrices are strictly diagonally dominant matrices, and the Schur complements of Ostrowski matrices are Ostrowski matrices.These properties have been used for deriving matrix inequalities in matrix analysis and for the convergence of iterations in numerical analysis (see [9]- [12]).More importantly, studying the locations for the eigenvalues of the Schur complement is of great significance, as shown in [2] [6] [13]- [18].
The paper is organized as follows.In Section 2, we give some new estimates of diagonally dominant degree on the Schur complement of matrices.In Section 3, we present several new eigenvalue inclusion regions for the Schur complement of matrices.In Section 4, we give a numerical example to illustrate the advantages of our derived results.

The Diagonally Dominant Degree for the Schur Complement
In this section, we present several new estimates of diagonally, α-diagonally and product α-diagonally dominant degree on the Schur complement of matrices.
( ) is the complement of β in N and β is the cardinality of β .Lemma 4. [16] where ( ) From Lemma 1 and Lemma 2, we have .
Thus, for any > 0 ε and 1 t l ≤ ≤ , we obtain , , A then there exists sufficiently small positive number 0 Construct a positive diagonal matrix ( ) , where ( ) . For 1 p = , by (3), we have ( ) ( ) , and so . Note that ( ) Let x be , by (4), we have Then we obtain (1).Similarly, we can prove (2).□ Remark 1.Note that This shows that Theorem 1 improves Theorem 2 of [17] and [2], respectively.Next, we present some new estimates of α-diagonally and product α-diagonally dominant degree of the Schur complement. , where for any By Lemma 1 and Lemma 2, we have , , .
Similar as the proof of Theorem 1, we can prove Similarly, we have , , .
This shows that Theorem 3 improves Theorem 4 of [2].Similar as the proof of Theorem 2, we can prove the following theorem immediately, which improves Theorem 2 of [2].

Eigenvalue Inclusion Regions of the Schur Complement
In this section, based on these derived results in Section 2, we present new eigenvalue inclusion regions for the Schur complement of matrices.Theorem 4. Let and λ be eigenvalue of β A .Then there exists 1 t l ≤ ≤ such that ( ) ( ).
Proof.By Lemma 5, we know that there exists 1 t l ≤ ≤ such that Similar as the proof of Theorem 2, we can prove .

A Numerical Example
In this section, we present a numerical example to illustrate the advantages of our derived results.Example 1.Let