1. Introduction
Throughout the paper, let n be an integer number, be the set of all indices, (resp.) denote the set of all complex (resp.real) matrices, be an identity matrix, , denote deleted ith row sum, and .
Definition 1 ([
1]).
Let (), then A is called:- (i)
A row diagonally dominant () if for all , - (ii)
A strictly diagonally dominant () if all the strict inequalities in (1) hold;
- (iii)
A doubly strictly diagonally dominant () matrix, if:
The class of generalized doubly strictly diagonally dominant (
GDSDD) matrices was presented by Gao and Wang in [
2].
Definition 2 ([
2]).
A matrix A is called a GDSDD
matrix if and there exist proper subsets of N such that and:for any and , where with or j, Here the and may be interpreted as the sums of the absolute values of the nondiagonal elements in row s that fall in the columns and , respectively. When , a GDSDD matrix is nothing but a matrix. A matrix A satisfying (3) may not be generalized doubly diagonally dominant for another pair subsets and . Assuming that the matrix order is , we adopt the notation: for generalized doubly strictly diagonally dominant.
Definition 3 ([
3]).
A matrix A is called an H-matrix if its comparison matrix defined by:is an M-matrix, i.e., .
It is shown in [
3] that if
A is an
H-matrix, then:
In addition, it was shown that
matrices,
matrices, and
GDSDD matrices are subclass of
H-matrices [
4].
Upper bounds for the infinity norm of the inverse of nonsingular matrices can be used to the convergence analysis of matrix splitting and matrix multi-splitting iterative methods for solving the large sparse of linear equations. Moreover, the upper bounds can also be applied to the smallest singular value of matrices [
5,
6].
One traditional method for finding upper bounds for the infinity norm of the inverse of nonsingular matrices is to use the definition and properties of a given matrix class; see [
7,
8,
9,
10] for details. The first work may due to Varah [
10], who in 1975 presented the following upper bound for the infinity norm of the inverse for
matrices.
Theorem 1 ([
10]).
If is , then: However, if the diagonal dominance of
A (i.e.
) is weak, Varah’s bound may yield a large value. In 2020, based on the Schur complement, Li [
11] obtained two upper bounds for the infinity norm of inverse of
matrices.
Using the definition and properties of
matrices, Liu, Zhang and Liu in [
9] obtained an upper bound of
for a
matrix
A as follows.
Theorem 2 ([
7]).
If is , then: In 2020, based on the Schur complement, Sang [
12] obtained two upper bounds for the infinity norm of
matrices.
And Moraa in [
13] give an upper bound for the infinity norm of
GDSDD matrices as follows.
Theorem 3. If is , then: In this paper, based on the Schur complement, we present some upper bounds for the infinity norm of the inverse of GDSDD matrices, and numerical examples are given to show the effectiveness of the obtained results. In addition, applying these new bounds, a lower bound for the smallest singular value of GDSDD matrices is obtained.
2. Schur Complement-Based Infinity Bounds for the Inverse of GDSDD Matrices
Firstly, we recall the Schur complement of matrices.
Let
be a proper subset of
N and denote by
the cardinality of
and by
the complement to
in
N. For nonempty index sets
, we write
to mean the submatrix of
lying in the rows indexed by
and the columns indexed by
.
is abbreviated to
. Throughout this paper, supposing
, the complement
and the elements of
and
are both conventionally arranged in increasing order. Furthermore, if
is nonsingular, we define the Schur complement of
A with respect to
, which is denoted by
or simply
, to be:
Liu, Huang, and Zhang in [
14] proved that the Schur complements of
GDSDD matrices are
GDSDD matrices. Many similar or related results had been obtained; see [
9,
15,
16] for details.
For a given nonempty proper subset
, there is always a permutation matrix
P such that:
It is well known that the inverse of a permutation matrix is also a permutation matrix and the infinity norm for the inverse of a permutation matrix equals to 1 (
). Hence, if
A is nonsingular, then:
So we next only consider the upper bound for
. For the sake of simplicity, we consider that:
where
and
.
Lemma 1 ([
17]).
Let with the form (6), and (resp. ) be the identity matrix of order l (resp. m). If is nonsingular, then:where From Lemma 1, we obtain:
and hence:
which implies that:
This implies that if upper bounds for
,
,
, and
are given, then the product of these bounds could be regarded as an upper bound for
by (7). It is not difficult to compute that:
and
In 2020, Li [
11] gave an upper bound for
as follows:
Lemma 2 ([
11]).
Let be nonsingular with for , and . If is nonsingular, and: Next, we only consider upper bounds for , and when A is .
Lemma 3 ([
14]).
Let A be . Then and are . Lemma 4 ([
14]).
Let A be . Then or . In this paper, we assume .
Lemma 5 ([
14]).
Let A be and . Then is . Lemma 6 ([
14]).
Let A be . Then for any proper subset α of N, is . Lemma 7 ([
18]).
Let be , and . Then:where . Lemma 8. Let be , and . Then:where . Proof. Since there exists an m-dimensional vector
such that
and:
where
, we have
, and let
, which implies that:
The proof is divided into two cases.
Case 1:
. Then,
yields
Case 2:
. Then,
yields:
The conclusion follows from equality (11) and inequalities (14) and (15). □
Corollary 1. Let be , and . Then:where Specially, if or , then is , and For the sake of convenience, denote:
for any
. And thus:
Lemma 9. Let , . If is a strictly diagonally dominant row, then: Proof. For any
,
is GDSDD, so is
, and consequently,
is an H-matrix, and
by (4). Hence, x is nonnegative. Let
. By
is a strictly diagonally dominant row, then:
By the gth equality of (20) and inequality (21), we have:
The proof is completed. □
Theorem 4. Let be with , and . Assume that (8) holds, if , then:where Proof. Denote the
-entry of
by
. Then for any
,
and by inequality (4), we have:
Since
A is
,
, by Lemmas 5 and 6, we have
is
, and
is
. Applying the Varah’s bound to
, we get:
Applying Theorem 3 to
yields:
By Theorem 2 in [
14], we have:
Now, the upper bounds of and are considered. Since , it satisfies Lemma 9, which implies that (19) holds.
By (19), (20), (23) and (24), we have:
Furthermore, by (26)–(29), we have:
where
Furthermore, by (7), (9), (25), (30), and (31), the conclusion follows. □
Theorem 5. Let be with , and . Assume that (8) and (19) hold, if , then:where Proof. Since
A is
,
, by Lemmas 5 and 6, we have
is
, and
is
. Applying the Varah’s bound to
, we get:
Applying Theorem 3 to
yields:
By Theorem 2 in [
14], we have:
Now, the upper bounds of and are considered.
By (19), (20), (23), and (24), we have:
Furthermore, by (33)–(36), we have:
where
Furthermore, by (7), (9), (32), (37), and (38), the conclusion follows. □
Theorem 6. Let be with , and . Assume that (8) and (19) hold, if , and , then:where Proof. Since
A is
,
and
, by Lemmas 5 and 6, we have
is
, and
is
. Applying Theorem 3 to
, we get:
Applying Theorem 3 to
yields:
By Theorem 2 in [
14], we have:
Now, the upper bounds of and are considered.
By (19), (20), (23), and (24), we have:
Furthermore, by (40)–(43), we have:
where
Furthermore, by (7), (9), (39), (44), and (45), the conclusion follows. □
Theorem 7. Let be with , and . Assume that (8) holds, if or , then:where Proof. Since
A is
,
or
, by Lemmas 5 and 6, we have
is
and
is
, consequently, nonsingular. Let:
Then by Theorem 1 in [
14], we have:
Applying the Varah’s bound to
, we get:
where
From (7), (9), (46), (47), and Corollary 1, the conclusion follows. □
Theorem 8. Let be with , and . Assume that (8) holds, if or , then:where Proof. Since
A is
,
or
, by Lemmas 5 and 6, we have
and
is
, consequently, nonsingular. Applying the Varah’s bound to
, we get:
From Corollary 1, we can obtain:
Then by Theorem 1 in [
14], we have:
Applying the Varah’s bound to
, we get:
where
From (7), (9), (48)–(50), the conclusion follows. □
Theorem 9. Let be , and . Then:where Proof. Since
A is GDSDD, then for any
, by Lemmas 5 and 6, we have
is
,
is
, and consequently, they are nonsingular. Taking
, then
,
and:
Let
. By (22), we have:
By calculation, we obtain for
,
By (7), (51)–(54), we have . By the arbitrariness of i, the conclusion follows. □
In the following, we prove that the bound in Theorem 9 generally improves the bound obtained by Theorem 3 in [
11] for
matrices and Theorem 9 in [
12] for
matrices.
Theorem 10. Let be . Then,where is defined in Theorem 9 and: Proof. Since
A is
,
A is GDSDD with
. From Theorem 9, we can obtain:
where
From Theorem 3 of [
11],
where
For all
, it is easy to get:
The proof is completed. □
Theorem 10 shows that the bound in Theorem 9 generally improves the bound obtained by Theorem 3 in [
11] for
matrices.
Theorem 11. Let be . Then:where is defined in Theorem 9, Proof. Since
A is
, there is at most one
such that
. If there does not exist an
such that
, then A is
, the proof is the same as Theorem 10. If there exists only one
such that
, then A is
GDSDD with
. From Theorem 9, we can obtain:
where
From Theorem 9 of [
12],
where
The proof is completed. □
Theorem 11 shows that the bound in Theorem 9 generally improves the bound obtained by Theorem 9 in [
12] for
matrices.
We illustrate our results by the following two examples.
Example 1. Consider the bound for of a matrix A, where: By Theorem 3 in [11], we have: By Theorem 9 in [12], we have: In fact, . This example shows that our bound is better than those in Theorems 1–3 and Theorem 3 in [11], Theorem 9. Example 2. Consider the bound for of a GDSDD
matrix A for , We know that A is neither a matrix nor a matrix, and the bound of can not be obtained by Theorems 1 and 2, Theorem 3 in [11], or Theorem 9 in [12], but it can be estimated by Theorems 3 and 9 in this paper. The bound for can be estimated by Theorem 9 as follows: 3. Applications to the Smallest Singular Value for GDSDD Matrices
The singular values of an
complex matrix
A are the eigenvalues of
and are denoted:
The smallest singular value plays a special role in expressing properties of matrices. It indicates not only whether
A is nonsingular, but also how far from the singular matrices
A is. In addition, the spectral condition number
is important in studying numerical calculations involving
A [
6]. Hence, lower bounds for the smallest singular value of matrices are of interest.
For
matrices case, Varah in [
10] obtained a lower bound for
which is listed as follows.
Theorem 12 ([
10]).
If a matrix and its transpose are all , then: Besides the lower bound in Theorem 12, there are other existing lower bounds for
, see [
5,
6,
11] and references therein. Based on Theorem 9, we can get a new lower bound for
.
Theorem 13. If a matrix and its transpose are all GDSDD
, then:where is defined as in Theorem 9. Proof. Since
is
GDSDD, then:
By the well-known inequality (see Theorem 2 in [
5]):
By , the conclusion follows. □