The disc separation and the eigenvalue distribution of the Schur complement of nonstrictly diagonally dominant matrices

The result on the Geršgorin disc separation from the origin for strictly diagonally dominant matrices and their Schur complements in (Liu and Zhang in SIAM J. Matrix Anal. Appl. 27(3):665-674, 2005) is extended to nonstrictly diagonally dominant matrices and their Schur complements, showing that under some conditions the separation of the Schur complement of a nonstrictly diagonally dominant matrix is greater than that of the original grand matrix. As an application, the eigenvalue distribution of the Schur complement is discussed for nonstrictly diagonally dominant matrices to derive some significant conclusions. Finally, some examples are provided to show the effectiveness of theoretical results.


Introduction
Let A = (a ij ) ∈ C n×n with R i (A) = n j=,j =i |a ij | for all i ∈ n define the disc separation of A by the quantities |a ii | -R i (A) that measure the separations of the discs |za ii | ≤ R i (A), i = , , . . . , n, from the origin and give estimate min ≤i≤n |a ii | -R i (A) of the absolute value of the shortest eigenvalue (the eigenvalue with the smallest absolute value, see []) of the strictly diagonally dominant matrix A ∈ C n×n .
In , Liu and Zhang [] firstly studied the disc separation of the Schur complements of strictly diagonally dominant matrices. More precisely, they compared a disc separation of the Schur complement to that of the original matrix and showed that each Geršgorin disc of the Schur complement is paired with a particular Geršgorin disc of the original matrix; the latter is further from the origin than the former. Their result is as follows.
On the other hand, Liu and Zhang [] also improved the result of Theorem  in [] when A ∈ SD n ⊂ H S n and established the new result on the eigenvalue distribution for the Schur complements of strictly diagonally dominant matrices with real diagonal entries.
Theorem  (see []) Let A = (a ij ) be an n × n strictly diagonally dominant matrix with real diagonal entries, and α ⊂ n . Then A/α and A(α ) have the same number of eigenvalues whose real parts are greater (less) than w (resp. -w), where α = nα and In this paper, we will generalize the result on the Geršgorin disc separation from the origin for strictly diagonally dominant matrices and their Schur complements in [] to nonstrictly diagonally dominant matrices and their Schur complements, showing that under some conditions the separation of the Schur complement of a nonstrictly diagonally dominant matrix is greater than that of the original grand matrix. As an application, we continue discussing the eigenvalue distribution of the Schur complements for nonstrictly diagonally dominant matrices to derive some significant conclusions.
The paper is organized as follows. Some notations and preliminary results about nonstrictly diagonally dominant matrices are given in Section . Some results on the Geršgorin disc separation from the origin are established in Section  for nonstrictly diagonally dominant matrices and their Schur complements. The eigenvalue distribution of the Schur complements is discussed in Section  for nonstrictly diagonally dominant matrices to derive some significant conclusions. Some examples are provided in Section  to show the effectiveness of theoretical results. Conclusions are given in Section .

Preliminaries
In this section we give some notions and preliminary results about special matrices that are used in this paper.
C m×n (R m×n ) will be used to denote the set of all m × n complex (real) matrices. Z denotes the set of all integers. Let α ⊆ n = {, , . . . , n} ⊂ Z. |α| denotes the cardinality of the set α. For nonempty index sets α, β ⊆ n , A(α, β) is the submatrix of A ∈ C n×n with row indices in α and column indices in β. The submatrix A(α, α) is abbreviated to A(α).
Let A ∈ C n×n , α ⊂ n and α = nα. If A(α) is nonsingular, the matrix is called the Schur complement with respect to A(α), indices in both α and α are arranged with increasing order. We shall confine ourselves to the nonsingular A(α) as far as A/α is concerned.
We will use Z n to denote the set of all n × n Z-matrices. A matrix A = (a ij ) ∈ Z n is called an M-matrix if A can be expressed in the form A = sI -B, where B ≥ , and s ≥ ρ(B), the spectral radius of B. If s > ρ(B), A is called a nonsingular M-matrix M n and M • n will be used to denote the set of all n × n M-matrices and the set of all n × n nonsingular M-matrices, respectively (see []).
The comparison matrix of a given matrix A = (a ij ) ∈ C n×n , denoted by μ(A) = (μ ij ), is defined by H n and H I n will denote the set of all n × n general H-matrices and the set of all n × n invertible H-matrices, respectively (see []).
For n ≥ , an n × n complex matrix A is reducible if there exists an n × n permutation matrix P such that

If no such permutation matrix exists, then A is called irreducible. If A is a  ×  complex matrix, then A is irreducible if its single entry is nonzero, and reducible otherwise.
holds for all i ∈ n . If inequality in () holds strictly for all i ∈ n , A is called strictly diagonally dominant by row. If A is irreducible and the inequality in () holds strictly for at least one i ∈ n , A is called irreducibly diagonally dominant by row. If () holds with equality for all i ∈ n , A is called diagonally equipotent by row. If () holds with equality for at least one i ∈ n , A is called nonstrictly diagonally dominant.
D n (SD n , ID n ) and DE n will be used to denote the sets of all n × n (strictly, irreducibly) diagonally dominant matrices and the set of all n × n diagonally equipotent matrices, respectively.
where A  = (a  , a  , . . . , a n ) T and A  = (a  , a  , . . . , a n ). If A  is nonsingular, then

The disc separation of the Schur complement of nonstrictly diagonally dominant matrices
In this section, we will establish some results on the Geršgorin disc separation from the origin for nonstrictly diagonally dominant matrices and their Schur complements such that under some conditions the separation of the Schur complement of a nonstrictly diagonally dominant matrix is greater than that of the original grand matrix. Firstly, the following lemma will be used in the rest of this subsection.

Then B jt is doubly diagonally dominant [] if and only if
for all j t ∈ α , then both () and () hold.
Proof Since A ∈ D n and is nonsingular, Lemma  indicates that A(α) is nonsingular. As a result, A/α = ( a j t ,j s ) exists. According to definition () of the Schur complement matrix A/α, we have the off-diagonal entries and the diagonal entries a j l ,j l = a j l ,j l - of A/α. The conclusion of this theorem will be proved by proving the following two cases: Then from (), (), () and Lemma , we have where It is clear that B j t ∈ Z m+ . In Lemma , we set This completes the proof of case (i). Next, we prove case (ii). Assume A(α) / ∈ H I |α| , it then follows from Lemma  that A(α) has at least one diagonally equipotent principal submatrix. Let A(γ ) be the largest diagonally equipotent principal submatrix of the matrix A(α) for γ = αγ ⊆ α. Then A(γ ) has no diagonally equipotent principal submatrix and hence A(γ ) ∈ H I |γ | from Lemma . Since A ∈ D n and A(γ ) is the largest diagonally equipotent principal submatrix of the matrix A(α), it follows from Lemma  that A/α = A(α ∪ γ )/γ , where α = nα ⊆ n and A(α ∪ γ ) is given in (). Since A(γ ) ∈ H I |γ | ∩ D |γ | , it follows from the proof of case (i) that both () and () hold, which shows that the proof of case (ii) is completed. This completes the proof.
Theorem  Given a matrix A = (a ij ) ∈ D n and two sets α = {i  , i  , . . . , i m } ⊂ n and α = nα = {j  , j  , . . . , j l } ⊂ n with m + l = n, define ω j t as in () and and Proof Using the same proof method as the one of Theorem , the conclusion of this theorem is obtained immediately.
Theorem  Given a matrix A = (a ij ) ∈ D n and two sets α = {i  , i  , . . . , i m } ⊂ n and α = nα = {j  , j  , . . . , j l } ⊂ n with m + l = n, define ω j t as in (), if A(α) is nonsingular, then Proof Similar to the proof of Theorem , one may derive the conclusion of this theorem.

The eigenvalue distribution of the Schur complement of nonstrictly diagonally dominant matrices
In this section, the result of Theorem  will be generalized to nonstrictly diagonally dominant matrices.
Theorem  Let A = (a ij ) ∈ D n be nonsingular with real diagonal entries, and α ⊂ n such that for all j ∈ α = nα ⊂ n , |a jj | > R j (A) = n k=,k =j |a jk |. Then A/α and A(α ) have the same number of eigenvalues whose real parts are greater (less) than w (resp. -w), where w is defined in ().
Proof Since A ∈ D n and is nonsingular, it follows from Lemma  that A(α) is nonsingular.
then A/α and A(α ) have the same number of eigenvalues whose real parts are greater (less) than ϑ (resp. -ϑ).
Proof Since A ∈ D n and ϑ = min i∈α,j∈α [ |a ii |-R i (A) |a ii | k∈α |a jk |] > , |a ii | > R i (A) for all i ∈ α and A(α ) has no diagonally equipotent principal submatrix. Then it follows from Lemma  that A is nonsingular, so is A(α). Thus, A/α exists. Similar to the proof of Theorem , the conclusion of this theorem is derived immediately.

Numerical examples
In this section, some numerical examples are given to demonstrate the effectiveness of the results obtained in this paper.
Example  Consider the following  ×  matrix: In what follows we will verify the conclusions of Theorems  and . Direct computations yield the following results: | a , | + R  (A/α) ≤ |a , | + R  (A)ω  ≤ |a , | + R  (A).
() and () show that the conclusions of Theorem  are true. It follows that we will verify the effectiveness of Theorem . Direct computations give ϑ = . > . According to Theorem , A/α and A(α ) have the same number of eigenvalues whose real parts are greater (less) than ϑ (resp. -ϑ).
By direct computations, eigenvalues of A/α and A(α ) are -., . + .i, . -.i and -., . + .i, . -.i, respectively. As a result, A/α and A(α ) have two eigenvalues whose real parts are greater than . and have an eigenvalue whose real part is less than -., respectively. This shows that the conclusion of Theorem  is true.

Conclusions
This mainly studies the disc separation and the eigenvalue location of some special matrices. Firstly, the result on the Geršgorin disc separation from the origin for strictly diagonally dominant matrices and their Schur complements in [] is extended to nonstrictly diagonally dominant matrices and their Schur complements to reveal that under some conditions the separation of the Schur complement of a nonstrictly diagonally dominant matrix is greater than that of the original grand matrix. Secondly, some significant conclusions are derived to establish the eigenvalue distribution of the Schur complements for nonstrictly diagonally dominant matrices. Finally, some examples are provided to demonstrate the effectiveness of some theoretical results obtained in this paper.