On Classes of Matrices with Variants of the Diagonal Dominance Property ()
1. 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
matrices over the field
of complex numbers by
. For every
and every
, we define the row sum
and column sum
by
(1.1)
In (1.1), it is understood that
if A is a
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
and
are subclasses of
, we say that
is a pair of incomparable classes if
and
.
2) If
is a pair of incomparable classes of matrices in
with variants of the diagonal dominance property, provide sufficient conditions for matrices in
to be in
, where
. We investigate this problem for most pairs of incomparable classes identified in 1).
The set of positive integers is denoted by
, and for every
, we denote the set
by
. The empty set is denoted by
. We denote the cardinality of a nonempty finite set S by
. The set of all
complex matrices is denoted by
. The set
is simply written as
. If
and
is the entry of x in the ith row,
, we write x as
. We denote by 0 the zero matrix, and when there is a need to emphasize its size, we will use the symbol
to denote the
zero matrix in
. The multiplicative group of
invertible matrices is denoted by
, and its identity is written as
. Let
. The entry
of A is sometimes written as
. The transpose of A is denoted by
. If
(
) for all
and
, we write
(
). The matrix
is the matrix in
defined by
for all
and
. If
is diagonal and
is the ith diagonal entry of
,
, we denote
by
. If
, the diagonal matrix
is denoted by
.
The set of eigenvalues of
is denoted by
, and the spectral radius of A is written as
. So,
. Similar matrices in
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
such that
. The set of all matrices, which are diagonally similar to
, is denoted by
. If
, we define the diagonal similarity orbit
of
by
(1.2)
Submatrices play a role in the development of the topics studied in the paper. Let
,
, and let S and T be nonempty subsets of
. The submatrix of A that lies in the rows and columns of A indexed by S and T, respectively, is denoted by
. If
, we write
simply as
; see p. 17 of [26] . For every nonempty subset S of
and each
, it is instructive to evaluate the
-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
and
by
(1.3)
It is clear that
and
. The sums in (1.3) are used in association with the notion of separation of
.
Definition 1.1 Let
. If S is a nonempty proper subset of
, we call the pair
a separation of
.
Remark 1.1 Let
,
. Let S be a subset of
with
. If
, where
, then
,
and
for all
.
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
separation-induced doubly diagonally dominant matrices. We analyze in Section 5 the relation between the class of generalized diagonally dominant matrices and the class of
separation-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
and the other classes we considered in Section 2.
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
is called irreducible if it not reducible. A matrix
is called reducible if either
and
; or
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
by
.
Definition 2.1 Let
. Define the sets
and
by
(2.1)
1) The matrix A is called diagonally dominant if
for all
. If
, 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
such that
is diagonally dominant. We call A strictly generalized diagonally dominant (also known as nonsingular H-matrix; see [11] ) if there exists a nonsingular diagonal matrix
such that
. If there exists a nonsingular diagonal matrix
such that
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
(2.2)
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
be a separation of
. We say that A is
separation- induced doubly diagonally dominant if
(2.3)
for all
and
. A is called
separation-induced strictly doubly diagonally dominant if
is nonempty and the inequalities of (2.3) are strict for all
and
. We say that A is
separation-induced irreducibly doubly diagonally dominant if A is irreducible,
is nonempty, A is doubly diagonally dominant with respect to the separation
and there exist
and
such that
(2.4)
4) Let
. We call A row-column diagonally dominant with index
if
(2.5)
If all the inequalities in (2.5) are strict, we say that A is strictly row-column diagonally dominant with index
A is called irreducibly row-column diagonally dominant with index
if A is irreducible, row-column diagonally dominant with index
and there exists
such that
(2.6)
Let
To simplify the terminology, we introduce the following abbreviated notations:
(2.7)
(2.8)
In the following terminology, we assume
:
(2.9)
If
is a separation of
, we introduce the notation
(2.10)
If
, then
(2.11)
3. 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.
Remark 3.1 Let
. Then
1)
,
and
.
In items (2)-(6), we assume
:
2)
,
and
.
3)
.
4) If
is a separation of
, then
i)
, and similar equalities hold for
and
.
ii)
.
iii)
.
iv) If
, then
if and only if A is irreducible,
and there exist
and
such that (2.4) holds and
(3.1)
v)
and
.
5)
,
and
.
6) If
, then
i)
.
ii)
.
iii)
.
iv)
.
v)
.
7) Let
. The classes
,
and
depend on the separation
of
. For example, the irreducible matrix A defined by
![]()
satisfies
but
. Then from (ii) of item (4), we deduce that
and
, and from A being irreducible and (iii) of item (4), we see that
and
.
The following fact is less obvious than the inclusions in (v) of item (4) of Remark 3.1.
Lemma 3.1 Let
, and let
be a separation of
. Then
.
Proof. Let
. It follows from
and (v) of item (4) of Remark 3.1 that
. Then from
and
, we deduce that in order to show that
, it remains to show the existence of
and
such that (2.4) is satisfied. From
and
being a separation of
, there exists
such that
. Assume that
. The other case is proven similarly. From
and
being a separation of
, there exists
such that
. Thus from
,
, we get
![]()
Hence, with taking
and
, we obtain (2.4).
Using (1.2), the following lemma provides characterizations of the classes
,
and
. The lemma somehow justifies the use of the word “generalized” in the titles for the 3 classes. We omit the proof.
Lemma 3.2 Let
. Then
,
and
.
Additional facts about the classes in Definition 2.1 are outlined in the following lemma.
Lemma 3.3 Let
. Then:
1)
and
, and the two inclusions are proper.
2)
.
3)
and
for all
.
4)
.
5)
.
6)
.
7) If
is a separation of
, then
(3.2)
and
(3.3)
8)
.
9) If
, then
(3.4)
Remark 3.2 We make the following observations in regard to Lemma 3.3.
1) In item (1), the inclusion
was established by Taussky ( [23] , Theorem II).
2) The inclusion of item (2) is proper; it was first proved by Ostrowski [19] .
3) If
, the fact
in item 3) was illustrated through examples in [21] and [25] .
4) Items (5) and (6) follow through a careful reading of the proof of Pro-
position 1 of [18] . If
then
. If
, then
, where
is the unique integer in
satisfying
(see (2.1)). If
then
. If
, then
, where m is the unique integer in
satisfying
.
5) In contrast to items (5) and (6), we observe that
![]()
For example, let
. Then
, but
. Note that for the separation
of
, we have:
![]()
![]()
So,
![]()
Also, for the separation
of
, we have
. For the separation
of
, the strict inequality (2.4) is not satisfied, since
![]()
for
.
6) Gao and Wang ( [12] , Theorem 1) established (3.2). For every integer
, the inclusion is proper. We consider the following two cases:
Case 1:
. Define the matrix
by
(3.5)
Then, with
(3.6)
we have
. Thus
. However, it can be shown that for every separation
of
, there exists a pair
such that
![]()
Hence the matrix
defined by (3.5) satisfies
for any separation
of
.
Case 2:
. Define
by
(3.7)
where
is the matrix defined by (3.5). It then follows from (3.5), (3.6) and case 1 that the diagonal matrix
satisfies
. Thus
. Let
be a separation of
. From (3.5) and (3.7), it is clear that
![]()
and
![]()
So, to complete the proof that
, it remains to consider the case:
![]()
It then follows that
is a separation of
. Hence from (3.7), we deduce that
![]()
for all
. Then from case 1 and the fact that
is a separation of
, we see that there exist
and
such that
![]()
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
. In contrast to the inclusion in item (8), we observe that
(3.8)
For example, let
be defined by
and
for all
. Then
. It can be shown that
. We will use Theorems 4.1 and 5.2 to show that
for all
; 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
, it is assumed that
and the separation
of
satisfy the additional condition:
(3.9)
In general, matrices in
need not to satisfy (3.9); for example,
is in
, but
and
. 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 Theorem 5.2 to prove the first inclusion of (3.3); see Corollary 5.3.
10) In (3.4), Ostrowski [20] established the inclusion
, and Hadjidimos ( [15] , Theorem 2.1) proved the inclusion
. Item (4) of Theorem 6.6 provides a simple different proof of Hadjidimos’s result.
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
and
.
2) We will show in Theorem 5.1 that the relation between
and
is in contrast to the relation between
and
(given by (3.2)).
To simplify the set up of some statements in Sections 4 and 6, we introduce Definition 3.1.
Definition 3.1 Let
, and let
be a nonempty subclass of
. We say that
is invariant under the permutation similarity transformation if for every permutation matrix
and every
, we have
.
Remark 3.4 Let
,
and let
be a separation of
with
.
1) The classes
,
,
,
,
,
and
are all invariant under the permutation similarity transformation.
2) There exists a permutation matrix
such that the linear transfor- mation
defined by
is an isomorphism from
onto
.
Similar observations could be stated for the pairs:
![]()
4. Matrices with the Doubly Diagonal Dominance Property vs. Matrices with the
Separation-Induced Doubly Diagonal Dominance Property
We denote the Cartesian product of two nonempty sets X and Y by
.
Theorem 4.1 Let
, and let
be a separation of
. Then the elements of the set
![]()
are pairs of incomparable classes.
Proof. It follows from Remark 3.4 that it suffices to consider the case:
, where
. Also, from Remark 3.1 (item (3), and (iii) of item (4)), we see that it suffices to show the existence of
such that
(4.1)
and
(4.2)
We consider the following two cases:
Case 1:
. Assume without loss of generality that
. Define
as follows:
![]()
![]()
![]()
(4.3)
Define
by
![]()
![]()
(4.4)
Then A and B defined by (4.3) and (4.4) satisfy (4.1) and (4.2), respectively, in this case.
Case 2:
. Define
as follows:
![]()
![]()
(4.5)
Define
by
![]()
![]()
![]()
![]()
and
(4.6)
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
in the corollary is by virtue of item (5) of Remark 3.1.
Corollary 4.1 Let
. Then the inclusions
and ![]()
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
(
) to be in
(
), it suffices to provide such conditions for matrices in the smaller classes
(
). Also, from Lemma 3.1 and item (4) of Lemma 3.3, we see that in order to present sufficient conditions for matrices in
to be in
, it suffices to provide such conditions for matrices in the smaller class
. This provides the basis for the set-ups of Theorems 4.2-4.4.
2) Let
, and let
,
and
. Then there exist
such that
(see (2.1)),
and
. Suppose that
is a separation of
. Then
(4.7)
(4.8)
and
(4.9)
If
and
, then from
we obtain
, but this contradicts that
. The “not both” phrase in (4.7) follows from
and
. If
and
, then from
we get
, but this con- tradicts that
and
are disjoint. The “not both” phrase in (4.8) follows from
and
. The facts in (4.9) are proved similarly to the ones in (4.8). We will use (4.7), (4.8) and (4.9) in Theorems 4.2, 4.3 and 4.4, respectively.
Theorem 4.2 Let
,
, and let
be the integer such that
. Suppose that
is a separation of
such that
. In addition, assume that A satisfies one of the following two conditions:
Condition (1):
.
Condition (2):
for all
.
Then
.
Proof. Let
. Then from
, we get
(4.10)
and
(4.11)
where in (4.11), the first inequality follows from
and the second inequality follows from
. From (4.11), we see that if A satisfies either condition (1) or condition (2) then
. (Note that if A satisfies condition
(2) then, from
and
, we get
for all
.) Then from (4.10) and the fact that
was chosen arbitrarily, the result follows.
Theorem 4.3 Let
,
, and let
be the integer such that
. Suppose that
is a separation of
such that
. In addition, assume that
. Then
.
Proof. Let
. Then from
, we deduce that
![]()
and, from
and
, we get
![]()
This proves
.
Theorem 4.4 Let
,
. Let
be the integer such that
. Suppose that
is a separation of
such that
. In addition, assume that A satisfies the following two conditions:
Condition (1):
.
Condition (2): If
then there exists
such that
(4.12)
Then
.
Proof. It follows from
that
,
and
. From
and
(see item (2) of Remark 4.1), we deduce that
. Then from
, condition (1) and Theorem 4.2, we infer that
. So, it remains to show that there exist
and
such that (2.4) is satisfied. It follows from
and
that
(4.13)
If
, then, with the choice of any
and any
, we see from (4.13) that (2.4) is satisfied. Thus it remains to consider the case
. In this case, we deduce from condition (2) that there exists
such that (4.12) is satisfied. Hence from
(see condition (1)),
and
, we get
![]()
Then (2.4) is satisfied with
and
.
Theorem 4.5 provides sufficient conditions for matrices in the classes
and
to be in
and
, respec- tively. We prove item (2) of the theorem. Item (1) is proven similarly.
Theorem 4.5 Let
be a separation of
,
, and let
be such that the following two conditions are satisfied:
Condition (1):
for all
with
, and
for all
with
.
Condition (2):
for all
and
for all
.
Then
1) If
then
.
2) If
then
.
Proof. Assume that
. Then
![]()
for all
and
, and there exist
and
such that (2.4) is satisfied. Thus from condition (2), we deduce that
for all
and
, and
. Hence from condition (1) and
, we see that
.
Remark 4.2 1) In item (2) of Theorem 4.5, the condition
was not used to drive the result.
2) If
is a separation of
,
, and
, sufficient conditions for A to be in
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.
5. The Class
vs. the Classes
and ![]()
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
by
.
Theorem 5.1 Let
. Then
1) If
is a separation of
, then
.
2) If
, then
, and for every separa-
tion
of
, the pair
is a pair of incompar- able classes.
Proof. 1) Let
be a separation of
. Choose
and
, and define
by
and
for all
. Then
.
2) Let
. We construct
such that
for any separation
of
. The idea is to perturb the matrix defined in item (6) of Remark 3.2. Consider the following two cases:
Case 1:
. Define
by
(5.1)
Then, with
as defined by (3.6), we have
. Table 1 illustrates that
for any separation
of
.
Case 2:
. Define
by
(5.2)
where
is defined by (5.1). Thus, with
and
given by (3.6), we deduce from (5.1) that
. Hence
. Let
be a separation of
. Assume first that either
or
.
Assume without loss of generality that
. Choose
. Then from
, (5.1) and (5.2), we get
![]()
So, it remains to consider the case:
(5.3)
Since
is a separation of
and
, we infer from (5.3) that
is a separation of
. Thus from (5.2), we see that
![]()
for all
. Hence from case 1, we deduce there exist
and
such that
![]()
As
was chosen arbitrarily, we infer that
. For every separation
of
, the in-
comparability of the pair
follows from the first part and item (1).
The following theorem provides sufficient conditions for matrices in
to be in
.
Theorem 5.2 Let
, and let
be a separation of
. Suppose that
. In addition, assume that A satisfies the following two conditions:
Condition (1): There exists
such that
.
Condition (2): The sets
and
are nonempty.
Then
1) We have
(5.4)
and
(5.5)
2) We have
(5.6)
and for every
satisfying
(5.7)
the diagonal matrix
defined by
(5.8)
is nonsingular and satisfies
. In particular,
.
3) If there exist
and
such that (2.4) is satisfied, then either
or
, where
is the nonsingular dia-
gonal matrix defined by (5.7) and (5.8).
Proof. 1) It follows from
and condition (1) that the second inequality of (5.4) holds. Then from condition (2) and
, we deduce that there exists
such that
, and that the first inequality of (5.4) holds as well. Now, let
and
. It then follows from
and the definitions of
and
that
. Thus from (5.4), we get
and
, and from the definition of
, we obtain
. This completes the proof of (5.5).
2) It follows from
, condition (2) and (5.5) that (5.6) holds. Let
be a real satisfying (5.7), and define the diagonal matrix
by (5.8). Hence from
(in (5.6)) and
(in (5.7)), we infer that
is a diagonal matrix with positive diagonal entries. Also, from (5.8) and
, it is clear that
(5.9)
for all
and
, and
(5.10)
for all
. From the first strict inequality of (5.5), the definition of
(in (5.6)) and
(in (5.7)), we get
for all
. Then from the first equality of (5.9), and (5.10), we see that
(5.11)
for all
. Since
, we deduce from the definition of
(in (5.6)) and
(in (5.7)) that
for all
. Thus from the second equality of (5.9), and (5.10), we infer that
(5.12)
for all
. Since
and
for all
, we see from the first inequality of (5.4), (5.9) and (5.10) that
(5.13)
for all
. Since
for all
, we deduce from the second inequality of (5.4), (5.9) and (5.10) that
for all
. Hence from (5.11)-(5.13), we infer that
. Then from
being a nonsingular diagonal matrix, we see that
.
3) Assume that there exist
and
such that (2.4) is satisfied. We consider the following two cases:
Case 1:
and
. In this case, we have
. Thus from (2.4), (5.6) and (5.7), we deduce that
![]()
Hence from
, (5.9) and (5.10), the result follows.
Case 2: Either
or
. In this case, we have
(5.14)
Then from (2.4) and (5.4), we infer that
and
. Thus from
, (5.10) and (5.14), we see that either
or
. Hence from (5.9), the result follows.
Corollary 5.1 Let
. Then
1) For each separation
of
, we have
.
2)
.
Proof. 1) Let
be a separation of
, and let
be such that
. We show that A satisfies conditions (1) and (2) of Theorem 5.2. It follows from Remark 1.1 and
that condition (1) of Theorem 5.2 is satisfied. Also, from A being irreducible, we deduce that condition (2) of Theorem 5.2 is satisfied. Then from
and Theorem 5.2, we infer that
.
2) Let
. Thus there exists
such that
(5.15)
So, from
and item (1), the result follows if we can show that
. It follows from
that
and there exists
such that
. Hence from (5.15), we see that
, that is,
.
Remark 5.1 The irreducibility condition in Corollary 5.1 cannot be dropped.
Let
. Then A is reducible,
,
(as
) and
, but
.
If
in Theorem 5.2, we could relax condition (2) in the theorem.
Theorem 5.3 Let
,
, and let
. Assume that
,
and
. Then
.
Proof. Without loss of generality, assume that
. Define the diagonal matrix
by
![]()
Then
is nonsingular,
and
![]()
for all
. Thus from
, we deduce that
. Hence from Lemma 3.2, we see that
.
Remark 5.2 Let
, and let
be a separation of
. It follows from Theorem 4.1 (using the permutation similarity transformation technique) that there exists
. It is clear that B satisfies conditions (1) and (2) of Theorem 5.2 as well. So, from
and Theorem 5.2, we deduce that
. This obser- vation together with (3.8) lead to the following corollary.
Corollary 5.2 Let
. Then
is a pair of incom- parable classes.
The following corollary establishes the first inclusion of (3.3).
Corollary 5.3 Let
, and let
be a separation of
. Then
.
Proof. Let
. Then from (iv) of item (4) of Remark 3.1, we deduce that there exist
and
such that (2.4) and (3.1) are satisfied. Thus condition (1) of Theorem 5.2 is satisfied. Also, A satisfies condition (2) of Theorem 5.2 by virtue of being irreducible. Hence from
and item (2) of Theorem 5.2, we infer that
, where
is the nonsingular diagonal matrix defined by (5.7) and (5.8). Also, since
and
satisfy (2.4), we see from item (3) of Theorem 5.2 that
. Finally, from
and
being nonsingular diagonal matrix, we deduce that
. This completes the proof that
, that is,
.
It follows from (v) of item (4) of Remark 3.1 that in order to establish sufficient conditions for matrices in
,
, to be in
, it suffices to provide such conditions for matrices in
. In the following theorem, if a set is empty its maximum is understood to be 0.
Theorem 5.4 Let
,
, and let
be a separation of
. Let
be a diagonal matrix in
with positive diagonal entries such that
. Assume that A and
satisfy the following conditions:
Condition (1):
.
Condition (2):
for all
.
Condition (3): For all
, we have
and
.
Then
1) For every
, we have
.
2) If
and
, then
and
.
3) If A and
satisfy the additional condition:
Condition (4):
![]()
then
.
Proof. We first observe that the existence of the diagonal matrix
with positive diagonal entries, which satisfies
, is ensured by virtue of
and Lemma 3.2.
1) Let
. Since
, we deduce from condition (2) that
.
2) Let
, and assume that
. Then from condition (3),
and
, we infer that
and
.
3) Assume that A and
also satisfy condition (4). We first observe that the condition is logically viable by virtue of condition (2) and items (1) and (2). It follows from condition (1) that (2.3) is satisfied for all
and
. Also, from conditions (1) and (2), we infer that (2.3) is satisfied for all
and
with
. Thus, it remains to consider the case
and
with
. It follows from condition (3) and item (2) that
![]()
Hence from condition (4), we see that (2.3) also holds for all
and
with
.
Example 5.1 Let
, and let ![]()
and
. Then
and, with
, we see that
and conditions (1)-(4) of Theorem 5.4 are satisfied with
![]()
Then
.
6. Row-Column Diagonally Dominant Matrices with Index α vs. Matrices with Other Variants of the Diagonal Dominance Property
In this section, we investigate the relations between the class
and the other classes introduced in Definition 2.1. There has not been too much attention in the literature to discuss such relations.
For a matrix
,
, define the sets
,
,
,
and
by
(6.1)
It is clear that
is decomposed into the three mutually disjoint sets
,
and
. Theorem 6.2 investigates the relation between the classes
and
. A characterization of the class
is given in Theorem 5 of [27] . We will use the following slightly modified version of the result in Theorem 6.2.
Theorem 6.1 Let
,
. Define the sets
and
by (6.1). Then the following statements hold:
1) If A satisfies the condition
(6.2)
then
(6.3)
2)
for some
if and only if A satisfies condition (6.2) and the condition:
(6.4)
Also, if A satisfies conditions (6.2) and (6.4), then the reals
and
defined by
(6.5)
satisfy
and
for all
.
Theorem 6.2 Let
. Then
![]()
Proof. Let
. Then A satisfies condition (6.2). Also, from
and the definition of
(in (6.1)), we get
(6.6)
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
defined by the first equality of (6.5) satisfies
. Hence from the fact that A satisfies (6.2) and item (2) of Theorem 6.1, we see that
for all
, where
is the real defined by the second equality of (6.5).
Remark 6.1 There is no set inclusion between the classes
and
, or between the classes
and
as the one established in Theorem 6.2 between the classes
and
. However, Theorem 6.3 provides sufficient conditions for matrices in the classes
and
to be in the classes
and
, respectively.
We will use in Theorem 6.3 and other parts in the section the following remark.
Remark 6.2 Let
, and let
be the function defined by
for all
. Then
is continuous. Moreover,
1) If
, then
is strictly decreasing and
.
2) If
, then
is constant and
.
3) If
, then
is strictly increasing and
.
Theorem 6.3 Let
, and let
. Assume that A satisfies the following condition:
Condition (1): There exists a nonempty subset S of
such that for every
,
(6.7)
Then
1) For each
, we have
for all
.
2) If
and
then
.
3) If
and
then
.
Proof. 1) Let
, and let
. Since
![]()
we deduce from (6.7) that it remains to consider the case:
. In this case, we infer from Remark 6.2 that
and![]()
. Then
holds.
2) The result follows from item (1).
3) The result follows from item (1).
The following theorem discusses the relation between the classes
and
.
Theorem 6.4 Let
, and let
be a separation of
. Then
1)
.
2) We have
![]()
3) If
and
, then
,
,
and
.
Proof. It follows from Remark 3.4 that it suffices to prove the theorem in the case:
, where
.
1) Define
by
(6.8)
Then
. It can be shown by considering the cases
and
that
![]()
for all
and
. So,
. Also, from (6.8), we have
![]()
for all
. Then
.
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 that
. Let
be a matrix in
, which satisfies the following conditions:
(6.9)
(6.10)
and
(6.11)
The construction of B is possible by virtue of
, and
(6.12)
We observe that (6.12) follows from
, (6.9) and
. As
, we have
. From (6.10), it is clear that
. Also, it follows from (6.11) that
. This proves
. It then follows from Remark 3.1 ((ii) and (iii) of item (4), and (ii) of item (6)) that the remaining statements hold.
Remark 6.3 It is clear that the irreducible matrix A defined in item (1) of Theorem 6.4 satisfies
. Then from
, and (ii) and (iii) of item (4) of Remark 3.1, we see that the inclusions in (v) of item (4) of Remark 3.1 and Lemma 3.1 are all proper.
Let
. In Examples 6.1 and 6.2, we construct a matrix B which satisfies conditions (6.9)-(6.11). The case
is considered in Example 6.1, while the case
is considered in Example 6.2.
Example 6.1 Let
. Define
by
![]()
Then
, and
![]()
So,
. From (i) of item (4) of Remark 3.1, it is clear that
.
Example 6.2 Let
, and let
be such that
. Define
by
![]()
Then
, and
![]()
So,
.
Theorem 6.5 investigates the relations between the class
and each of the classes
and
.
Theorem 6.5 Let
. Then
1)
.
2)
,
and
.
3) If
and
, then
,
,
and
.
4) If
, then
.
Proof. 1) Define the matrix A by
(6.13)
and define
by
(6.14)
where A is given by (6.13) and
and
are given by
(6.15)
and
(6.16)
It follows from (6.13) - (6.16) that
, and with
defined by
![]()
we have
. Also, since
for all
, we deduce from (6.13)-(6.16) that
.
2) The statements follow from item (1), and items (1)-(3) and (6) of Remark 3.1.
3) Let
, and let
. Let
be a matrix in
, which satisfies the following condition:
,
. (Such matrix exists; see item (3) of Theorem 6.4.) Then
. It then follows from items (2), (3) and (6) of Remark 3.1 that the remaining statements also hold.
4) Let
. Define
by
and
for all
. Then
.
Remark 6.4 Let
. It is clear that the matrix
defined in item (1) of Theorem 6.5 satisfies
. Then from
,
and
, we see that the respective inclusions
,
and
in item (1) of Remark 3.1 are all proper.
Remark 6.5 We are not able to determine whether
or
. However, we show in Theorem 6.6 that a subclass of
containing
is indeed a subclass of
. The theorem also establishes
.
Definition 6.1 Let
, and let
. Define the class
by
(6.17)
For every
, define the function
by
(6.18)
for all
. Also, define the class
by
(6.19)
To simplify notation, we denote for every
, the matrix
by
.
Theorem 6.6 Let
, and let
. Then
1)
.
2) If
and
is a positive eigenvector of
, then the diagonal matrix
satisfies
(6.20)
for all
.
3)
,
and
, and the three inclusions are proper.
4)
.
Proof. 1) Let
. Then
for all
. Thus from (6.17), we deduce that
. Also, from
and (6.18), it is clear that
(see Definition 6.1) is both nonnegative and irreducible. Hence from Theorem 8.4.4 of [26] , we infer that
has a positive eigenvector. Then from
and (6.19), we see that
.
2) Let
, and let
be a positive eigenvector of
. Thus from
(see Definition 6.1) and Corollary 8.1.30 of [26] , we deduce that the eigenvalue of
corresponding to the positive eigenvector
of
is
. Hence from the definition of
, we get
(6.21)
for all
. It follows from
(in (3.4)) and (6.18) that for each matrix
, we have
(6.22)
Let
. Then from
(6.18) and the definition of
, we obtain
for all
. Thus from
being singular (as
is an eigenvalue of
) and (6.22), we infer that there exists
such that
. Hence from the definitions of
and
, we
see that
. Then from (6.18) and (6.21), we deduce that with
, inequality (6.20) holds for all
.
3) We prove
. The other two set inclusions are proven similarly. Let
. Thus from the definition of
, there exists a positive eigenvector
of
. Hence from item (2), we infer that, with
, inequality (6.20) is satisfied for all
. Then from
and
,
, we see that
. Thus from Lemma 3.2, we deduce that
.
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
such that
. Define
by
(6.23)
Hence
. Also, with
,
and
for all
, we deduce from (6.23) that
. So, from
, the result follows.
4) The result follows from
in item (3), and item (1) of Lemma 3.3.
We now provide sufficient conditions for matrices in the classes
,
and
to be in the classes
,
and
, respectively. It follows from Theorems 6.2 and 6.3 that in order to establish such conditions, it suffices to consider matrices in the smaller classes
,
and
. The integer n in Theorems 6.7-6.9 is assumed to satisfy
.
Theorem 6.7 Let
, and let l be the integer in
, which satisfies
. Assume that A satisfies the following two conditions:
Condition (1): Every
satisfies (6.7).
Condition (2): Either
, or
and
.
Then there exists
such that
.
Proof. Since
for all
, we deduce from condition (1) and item (1) of Theorem 6.2 that
(6.24)
for all
and
. It is clear that
(6.25)
Then from condition (2), it remains to consider the case in which l satisfies the conditions
and
. In this case, we infer from Remark 6.2 that there exist
and
such that
(6.26)
for all
. Since
, we see from Remark 6.2 that the function
,
, is an increasing function. Thus from (6.26) and
, we see that
for all
. Hence from (6.24) and (6.25), the result follows.
The following two theorems are proven similarly as Theorem 6.7.
Theorem 6.8 Let
, and let m be the integer in
, which satisfies
. Assume that A satisfies the following two condition:
Condition (1): Every
satisfies (6.7).
Condition (2): Either
, or
and
.
Then there exists
such that
.
It follows from item (4) of Lemma 3.3 that
. We use this fact in the following theorem. We omit the proof.
Theorem 6.9 Let
, and let
. Suppose that l is the integer in
, which satisfies
. Assume that A satisfies the following two conditions:
Condition (1): For every
, we have
.
Condition (2):
and
.
Then there exists
such that
![]()
In particular,
.
Remark 6.6 If
in Theorem 6.8, the fact:
for all
follows from
(as
).
Remark 6.7 1) Theorems 5.4 and 6.6 could be used to establish sufficient conditions for matrices in
to be 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
,
and
to be in
,
and
, respectively.
3) Theorems 4.5, 5.4 and 6.6 could be used to provide sufficient conditions for matrices in
to be in
.
Acknowledgements
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.