On Classes of Matrices with Variants of the Diagonal Dominance Property

We study the relations between several classes of matrices with variants of the diagonal dominance property, and identify those classes which form pairs of incomparable classes. For an incomparable pair ( ) 1 2 , X X of classes of matrices with variants of the diagonal dominance property, we also study the problem of providing sufficient conditions for the matrices in i X to be in j X with { } { } , 1, 2 i j = . The article is a continuation of a series of articles on the topic and related topics by the author; see [1] [2] [3] [4].


Introduction and Notation
The theory of matrices with variants of the diagonal dominance property has attracted the attention of researchers in matrix analysis and its applications.Desplanques [5] established the invertibility of every strictly diagonally dominant complex matrix; see Definition 2.1.(Lévy [6] established the result earlier for real matrices).The pioneering work of Lévy and Desplanques motivated researchers to study matrices with variants of the diagonal dominance property.
For more results on the subject; see, for example, [1] and [3]- [25].As usual, we denote the algebra of all n n × matrices over the field  of complex numbers by n n ×  .For every and .
n n i ij i ji j j i j j i r A a c A a In (1.1), it is understood that ( ) ( ) × matrix.The objectives of this paper are to investigate the following two problems: 1) Identify among several classes of matrices with variants of the diagonal dominance property those which form pairs of incomparable classes.If 1 X and 2 X are subclasses of n n ×  , we say that ( ) , X X is a pair of incomparable classes of matrices in n n ×  with variants of the diagonal dominance property, provide sufficient conditions for matrices in i X to be in j X , where { } { } , 1, 2 i j = . We investigate this problem for most pairs of incomparable classes identified in 1).
The set of positive integers is denoted by  , and for every n ∈  , we denote the set { } x is the entry of x in the ith row,

1, , i n
=  , we write x as ( ) 1 , , We denote by 0 the zero matrix, and when there is a need to emphasize its size, , and is diagonal .
Submatrices play a role in the development of the topics studied in the paper.Let The paper is organized as follows.In Section 2, we list the classes of matrices with variants of the diagonal dominance property, which we consider in the paper.Section 3 outlines some of the preliminary facts about the classes defined in Section 2. The section provides a motivation for the results in the remaining sections of the paper.In Section 4, we study in depth the relation between doubly diagonally dominant matrices and ( ) , S S separation-induced doubly diago- nally dominant matrices.We analyze in Section 5 the relation between the class of generalized diagonally dominant matrices and the class of ( ) , S S separa- tion-induced doubly diagonally dominant matrices.We also show that the former class forms with the class of doubly diagonally dominant matrices a pair of incomparable classes.In Section 6, we study the relations between the row-column diagonally dominant matrices with index α and the other classes we considered in Section 2.

Matrices with Variants of the Diagonal Dominance Property
We outline in this section the classes of matrices we consider in the rest of the paper.Irreducible matrices play an important role in the development of the theory.A matrix and B is similar by way of permutation to a strictly upper triangular block matrix; see Definition 6.2.21 in [26].We denote the set of all irreducible matrices in n n ×  by ( ) , we say that A is strictly diagonally dominant.We call A irreducibly diagonally dominant if ( ) and A is both diagonally dominant and irreducible.We say that A is generalized diagonally dominant if there exists a nonsingular diagonal matrix We call A strictly generalized diagonally dominant (also known as nonsingular H-matrix; see [11]) if there exists a nonsingular diagonal matrix . If there exists a nonsingular diagonal matrix that Aϒ is irreducibly diagonally dominant, we say that A is irreducibly generalized diagonally dominant.
In the following items, we assume 2) We call A doubly diagonally dominant if ( ) ( ) for all , with .
We say that A is strictly doubly diagonally dominant if the inequalities in (2.2) are all strict.If A is doubly diagonally dominant, irreducible and at least one of inequalities (2.2) is strict, we call A irreducibly doubly diagonally dominant.
3) Let ( ) , S S be a separation of n .We say that A is ( ) for all If all the inequalities in (2.5) are strict, we say that A is strictly row-column diagonally dominant with index .
α A is called irreducibly row-column diago- nally dominant with index α if A is irreducible, row-column diagonally do- minant with index α and there exists k n ∈ such that ( ) ( ) ( ) ( ) 1 .
n ∈  To simplify the terminology, we introduce the following abbreviated notations: : is strictly generalized diagonally dominant , : is irreducibly generalized diagonally dominant .
In the following terminology, we assume  If ( ) , S S is a separation of n , we introduce the notation

Preliminaries
Some of the important facts linking the classes introduced in Definition 2.1 are reviewed in this section.The information provide motivations for the results established in the subsequent sections.
In items ( 2)-( 6), we assume , S S is a separation of n , then i) ( ) , , , , S S n SDD and , , S S n IRDD depend on the separation ( ) , S S of n .For example, the irreducible matrix A defined by > .
Using (1.2), the following lemma provides characterizations of the classes . The lemma somehow justifies the use of the word "generalized" in the titles for the 3 classes.We omit the proof. ) . Additional facts about the classes in Definition 2.1 are outlined in the following lemma.
, and the two inclusions are proper.
4) Items ( 5) and ( 6) follow through a careful reading of the proof of Proposition 1 of [18].If ( ) , where l is the unique in- , where m is the unique integer in n satisfying

5)
In contrast to items ( 5) and ( 6), we observe that For example, let ( ) Gao and Wang ([12], Theorem 1) established (3.2).For every integer 5 n ≥ , the inclusion is proper.We consider the following two cases: Case 1: Then, with ( ) we have ( ) . However, it can be shown that for every separation ( ) , S S of 5 , there exists a pair ( ) where 5 A is the matrix defined by (3.5).It then follows from (3.5), (3.6) and case 1 that the diagonal matrix , S S be a separation of n .From (3.5) and (3.7), it is clear that ( ) ( ) and for all 5 m ∈ .Then from case 1 and the fact that ( )  is a separation of 5 , we see that there exist .
The integer 5 is the smallest integer we were able to find with which the inclusion of (3.2) is proper.
For example, let ( ) ( ) In general, matrices in , , S S n IRDD need not to satisfy (3.9); for example, . It is possible to establish (3.3) without making the assumption (3.9) by slightly modifying the proof of Theorem 2 of [12].However, we will use 2) We will show in Theorem 5.1 that the relation between , , S S n DD and , , S S n SDD and
To simplify the set up of some statements in Sections 4 and 6, we introduce Definition 3.1.
, and let  be a nonempty subclass of n n ×  .We say that  is invariant under the permutation similarity transformation if for every permutation matrix , S S be a separation of n with 1 cardS p = .
1) The classes ; n α SRCD and ( ) ; n α IRRCD are all invariant under the permutation similarity transformation.
2) There exists a permutation matrix such that the linear transformation ( ) ( ) , , S S n DD onto ( ) Similar observations could be stated for the pairs:    Proof.It follows from Remark 3.4 that it suffices to consider the case: , where 1 p n ∈ − .Also, from Remark 3.1 (item (3), and (iii) of item ( 4)), we see that it suffices to show the existence of , and We consider the following two cases: Case 1: { }  { }  The following corollary is a direct consequence of items ( 5) and ( 6) of Lemma ) to be in ( )

, , S S n SDD
), it suffices to provide such conditions for matrices in the smaller classes ( ) ( ) , S S is a separation of n .Then ( ) ( ) ( ) n ≥ , and let l n ∈ be the . In addition, assume that A satisfies one of the following two conditions: where in (4.11), the first inequality follows from ( ) , S S is a separation of n such that ( ) . In addition, assume that ( ) , , and, from ( )

( ) ( ) ( )
, S S is a separation of n such that ( ) . In addition, assume that A satisfies the following two conditions: , , A S S n ∈ IRDD .
Theorem 4.5 provides sufficient conditions for matrices in the classes , , S S n DD and , , S S n IRDD to be in , S S be a separation of n , 2 n ≥ , and let ∈  be such that the following two conditions are satisfied: Condition (1): , , A S S n ∈ DD then ( ) , , A S S n ∈ IRDD then ( ) , , A S S n ∈ IRDD .2) If ( ) , , A S S n ∈ SDD , sufficient conditions for A to be in ( ) n SDD could be set by replacing the inequalities in condition (1) of Theorem 4.5 by strict inequalities and keeping condition (2) of the theorem as it is.

The Class ( )
GD n vs. the Classes ( )

DD , , S S n and
( )

DD n
The first main result of this section is Theorem 5.1.In item (2) of the theorem, 5 is the smallest integer we were able to find, which satisfies the result.We denote the set of all separations of n by ( ) , S S is a separation of n , then , , , , , S S of n .The idea is to perturb the matrix defined in item (6) of Remark 3.2.Consider the following two cases: Case 1: Then, with 5 ϒ as defined by (3.6), we have ( ) , ,5 A S S ∉ DD for any separation ( ) , S S of 5 .Case 2: where 5 A is defined by (5.1).Thus, with

( )
, S S be a separation of n .Assume first that either Assume without loss of generality that  and for all 5 m ∈ .Hence from case 1, we deduce there exist .
, S S was chosen arbitrarily, we infer The following theorem provides sufficient conditions for matrices in , , S S n DD to be in ( ) , S S be a separation of n .Suppose that ( ) , , and for every γ satisfying . This completes the proof of (5.5).
Also, if A satisfies conditions (6.2) and (6.4), then the reals 1 α and 2 α de- fined by i n ∈  , we define the row sum ( ) i r A and column sum ( )

γ
we will use the symbol 0 n m × to denote the n m × zero matrix in n m ×  .The multiplicative group of n n × invertible matrices is denoted by The entry ij a of A is sometimes written as ( ) ijA .The transpose of A is denoted by t A is the ith diagonal entry of ϒ , have the same eigenvalues.Among similar matrices, those which are similar through diagonal matrices, are of particular interest.If , we say that A is diagonally similar to B if there exists a diagonal matrix

2 n
≥ , and let S and T be nonempty subsets of n .The submatrix of A that lies in the rows and columns of A indexed by S and T, respectively, is denoted by ( ) , A S T .If S T = , we write ( ) , A S S simply as ( ) A S ; see p.
17 of[26].For every nonempty subset S of n and each i n ∈ , it is instructive to evaluate the 1  -norm of the off-diagonal entries among the ith row (column), which belong to the columns (rows) of A defined by the set S. Formally, we define ( )

4 )Remark 3 . 2
We make the following observations in regard toLemma 3

3 n 2 ,
≥ ; see Remark 5.2.9) Theorem 2 of [12] establishes (3.3) through the two set inclusions.The second inclusion readily follows.In the proof of ∈ IRDD and the separation ( ) 1 S S of n satisfy the additional condition: We denote the Cartesian product of two nonempty sets X and Y by X Y be a separation of n .Then the elements of the set

3. 3 , and Theorem 4 . 1 . The exclusion of 2 n 1 )
= in the corollary is by virtue of item (5) of Remark 3It follows from (v) of item (4) of Remark 3.1 that in order to establish sufficient conditions for matrices in

1 S
= in Theorem 5.2, we could relax condition (2) in the theorem.

1 )
It is clear that n is decomposed into the three mutually disjoint sets 1, , n  by n .The empty set is denoted by ∅ .We denote the car- dinality of a nonempty finite set S by cardS .The set of all n m n x ∈  and i Farid ,