_{1}

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 (
*X*
_{1},
*X*
_{2}) of classes of matrices with variants of the diagonal dominance property, we also study the problem of providing sufficient conditions for the matrices in
*X*
_{i} to be in
*X*
_{j} with {i,j}={1,2}. The article is a continuation of a series of articles on the topic and related topics by the author; see [1][2][3][4].

The theory of matrices with variants of the diagonal dominance property has attracted the attention of researchers in matrix analysis and its applications. Desplanques [

In (1.1), it is understood that

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

2) If

The set of positive integers is denoted by

The set of eigenvalues of

Submatrices play a role in the development of the topics studied in the paper. Let

It is clear that

Definition 1.1 Let

Remark 1.1 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

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

Definition 2.1 Let

1) The matrix A is called diagonally dominant if

In the following items, we assume

2) We call A doubly diagonally dominant if

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

for all

4) Let

If all the inequalities in (2.5) are strict, we say that A is strictly row-column diagonally dominant with index

Let

In the following terminology, we assume

If

If

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.

Remark 3.1 Let

1)

In items (2)-(6), we assume

2)

3)

4) If

i)

ii)

iii)

iv) If

v)

5)

6) If

i)

ii)

iii)

iv)

v)

7) Let

satisfies

The following fact is less obvious than the inclusions in (v) of item (4) of Remark 3.1.

Lemma 3.1 Let

Proof. Let

Hence, with taking

Using (1.2), the following lemma provides characterizations of the classes

Lemma 3.2 Let

Additional facts about the classes in Definition 2.1 are outlined in the following lemma.

Lemma 3.3 Let

1)

2)

3)

4)

5)

6)

7) If

and

8)

9) If

Remark 3.2 We make the following observations in regard to Lemma 3.3.

1) In item (1), the inclusion

2) The inclusion of item (2) is proper; it was first proved by Ostrowski [

3) If

4) Items (5) and (6) follow through a careful reading of the proof of Pro-

position 1 of [

5) In contrast to items (5) and (6), we observe that

For example, let

So,

Also, for the separation

for

6) Gao and Wang ( [

Case 1:

Then, with

we have

Hence the matrix

Case 2:

where

and

So, to complete the proof that

It then follows that

for all

The integer 5 is the smallest integer we were able to find with which the inclusion of (3.2) is proper.

7) Item (8) follows from item (6) and (3.2).

8) Let

For example, let

9) Theorem 2 of [

In general, matrices in

10) In (3.4), Ostrowski [

Remark 3.3 1) In light of the facts given in items (5) and (6) of Lemma 3.3, we will analyze in more depth in Section 4 the relation between

2) We will show in Theorem 5.1 that the relation between

To simplify the set up of some statements in Sections 4 and 6, we introduce Definition 3.1.

Definition 3.1 Let

Remark 3.4 Let

1) The classes

2) There exists a permutation matrix

Similar observations could be stated for the pairs:

We denote the Cartesian product of two nonempty sets X and Y by

Theorem 4.1 Let

are pairs of incomparable classes.

Proof. It follows from Remark 3.4 that it suffices to consider the case:

and

We consider the following two cases:

Case 1:

Define

Then A and B defined by (4.3) and (4.4) satisfy (4.1) and (4.2), respectively, in this case.

Case 2:

Define

and

Then A and B defined by (4.5) and (4.6) satisfy (4.1) and (4.2), respectively, in this case.

The following corollary is a direct consequence of items (5) and (6) of Lemma 3.3, and Theorem 4.1. The exclusion of

Corollary 4.1 Let

are proper.

Remark 4.1 1) It follows from (v) of item (4) of Remark 3.1 that in order to establish sufficient conditions for matrices in

2) Let

and

If

Theorem 4.2 Let

Condition (1):

Condition (2):

Then

Proof. Let

and

where in (4.11), the first inequality follows from

(2) then, from

Theorem 4.3 Let

Proof. Let

and, from

This proves

Theorem 4.4 Let

Condition (1):

Condition (2): If

Then

Proof. It follows from

If

Then (2.4) is satisfied with

Theorem 4.5 provides sufficient conditions for matrices in the classes

Theorem 4.5 Let

Condition (1):

Condition (2):

Then

1) If

2) If

Proof. Assume that

for all

Remark 4.2 1) In item (2) of Theorem 4.5, the condition

2) If

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

Theorem 5.1 Let

1) If

2) If

tion

Proof. 1) Let

2) Let

Case 1:

Then, with

Case 2:

where

Separation of | Testing Pair | ||
---|---|---|---|

Assume without loss of generality that

So, it remains to consider the case:

Since

for all

As

comparability of the pair

The following theorem provides sufficient conditions for matrices in

Theorem 5.2 Let

Condition (1): There exists

Condition (2): The sets

Then

1) We have

and

2) We have

and for every

the diagonal matrix

is nonsingular and satisfies

3) If there exist

gonal matrix defined by (5.7) and (5.8).

Proof. 1) It follows from

2) It follows from

for all

for all

for all

for all

for all

3) Assume that there exist

Case 1:

Hence from

Case 2: Either

Then from (2.4) and (5.4), we infer that

Corollary 5.1 Let

1) For each separation

2)

Proof. 1) Let

2) Let

So, from

Remark 5.1 The irreducibility condition in Corollary 5.1 cannot be dropped.

Let

If

Theorem 5.3 Let

Proof. Without loss of generality, assume that

Then

for all

Remark 5.2 Let

Corollary 5.2 Let

The following corollary establishes the first inclusion of (3.3).

Corollary 5.3 Let

Proof. Let

It follows from (v) of item (4) of Remark 3.1 that in order to establish sufficient conditions for matrices in

Theorem 5.4 Let

Condition (1):

Condition (2):

Condition (3): For all

Then

1) For every

2) If

3) If A and

Condition (4):

then

Proof. We first observe that the existence of the diagonal matrix

1) Let

2) Let

3) Assume that A and

Hence from condition (4), we see that (2.3) also holds for all

Example 5.1 Let

and

Then

In this section, we investigate the relations between the class

For a matrix

It is clear that

Theorem 6.1 Let

1) If A satisfies the condition

then

2)

Also, if A satisfies conditions (6.2) and (6.4), then the reals

satisfy

Theorem 6.2 Let

Proof. Let

As A satisfies (6.2), we deduce from item (1) of Theorem 6.1 that A satisfies (6.3). Thus from (6.6), we infer that A satisfies (6.4) and the real

Remark 6.1 There is no set inclusion between the classes

We will use in Theorem 6.3 and other parts in the section the following remark.

Remark 6.2 Let

1) If

2) If

3) If

Theorem 6.3 Let

Condition (1): There exists a nonempty subset S of

Then

1) For each

2) If

3) If

Proof. 1) Let

we deduce from (6.7) that it remains to consider the case:

2) The result follows from item (1).

3) The result follows from item (1).

The following theorem discusses the relation between the classes

Theorem 6.4 Let

1)

2) We have

3) If

Proof. It follows from Remark 3.4 that it suffices to prove the theorem in the case:

1) Define

Then

for all

for all

2) The result follows from item (1), and Remark 3.1 ((iii) of item (4), and (i) and (ii) of item (6)).

3) Assume that

and

The construction of B is possible by virtue of

We observe that (6.12) follows from

Remark 6.3 It is clear that the irreducible matrix A defined in item (1) of Theorem 6.4 satisfies

Let

Example 6.1 Let

Then

So,

Example 6.2 Let

Then

So,

Theorem 6.5 investigates the relations between the class

Theorem 6.5 Let

1)

2)

3) If

4) If

Proof. 1) Define the matrix A by

and define

where A is given by (6.13) and

and

It follows from (6.13) - (6.16) that

we have

2) The statements follow from item (1), and items (1)-(3) and (6) of Remark 3.1.

3) Let

4) Let

Remark 6.4 Let

Remark 6.5 We are not able to determine whether

Definition 6.1 Let

For every

for all

To simplify notation, we denote for every

Theorem 6.6 Let

1)

2) If

for all

3)

4)

Proof. 1) Let

2) Let

for all

Let

(6.18) and the definition of

see that

3) We prove

It follows from Remark 3.1 (item (1), and (ii) of item (6)) that in order to show that the three set inclusions are proper, it suffices to show that there exists

Hence

4) The result follows from

We now provide sufficient conditions for matrices in the classes

Theorem 6.7 Let

Condition (1): Every

Condition (2): Either

Then there exists

Proof. Since

for all

Then from condition (2), it remains to consider the case in which l satisfies the conditions

for all

The following two theorems are proven similarly as Theorem 6.7.

Theorem 6.8 Let

Condition (1): Every

Condition (2): Either

Then there exists

It follows from item (4) of Lemma 3.3 that

Theorem 6.9 Let

Condition (1): For every

Condition (2):

Then there exists

In particular,

Remark 6.6 If

Remark 6.7 1) Theorems 5.4 and 6.6 could be used to establish sufficient conditions for matrices in

2) Theorem 4.5, item (2) of Remark 4.2, and Theorems 6.7-6.9 could be used to present sufficient conditions for matrices in

3) Theorems 4.5, 5.4 and 6.6 could be used to provide sufficient conditions for matrices in

The author would like to thank the two referees for their helpful suggestions. The author would also like to thank the library of UBC Okanagan for the research facilities they provided.

Farid, F.O. (2017) On Classes of Matrices with Variants of the Diagonal Dominance Property. Advances in Linear Algebra & Matrix Theory, 7, 37- 65. https://doi.org/10.4236/alamt.2017.72005