Some New Results on Determinantal Inequalities and Applications

Abstract

Some new upper and lower bounds on determinants are presented for diagonally dominant matrices and general -matrices by using different methods. These bounds are some improvements of results given by Ostrowski (1952) and (1937), Price (1951), Wang and Zhang (2002), Huang and Liu (2005), and so forth. In addition, these bounds are also used to localize some numerical characters (e.g., the minimum eigenvalues, singular values and condition numbers) of certain matrices.

1. Introduction

As it is well known, the determinant has a long history of application, which can be traced back to Leibniz (1646–1716), and its properties were developed by Vandermonde (1735–1796), Laplace (1749–1827), Cauchy (1789–1857) Jacobi (1804–1851), and so forth; see [1]. So it has hitherto great influence on every branch of mathematics (see, e.g., [1–4]).

Throughout the paper, let () denote the set of all complex (real) matrices and . For and any , we define

(1.1)

According to [5, 6], a matrix is called a diagonally dominant() one, if for any , , and a square matrix is strictly (row) diagonally dominant() if for each . A nonsingular -matrix [7] is a -matrix (i.e., all its off-diagonal elements are nonpositive) with nonnegative inverse, and a matrix is an -matrix () if and only if its comparison matrix is a nonsingular -matrix, where

(1.2)

-matrices and -matrices have an important role in many fields; see, for example, [5–7].

In addition, there are various generalizations of class. Recall that a doubly strictly diagonally dominant() [8] is a matrix such that for all , one has

(1.3)

If there exists a positive diagonal matrix such that , then is generalized strictly diagonally dominant(). A matrix is nothing but an -matrix (see [7, page 185]).

The estimation for determinants () is an attractive topic in matrix theory and numerical analysis, especially in mathematical physics, since computers are not very valid for analysis of matrices with parameters, which plays an essential role in various applications (see, [1–3]). Therefore, this problem has been discussed by many articles and some elegant and useful results were obtained as follows.

First, Ostrowski [9] proved, under the hypothesis , that

(1.4)

Subsequently, Price [10] suggested another new expression as

(1.5)

In [11], the above inequalities (1.4)-(1.5) are improved in such a way, for an arbitrary index , that

(1.6)

where with and representing and , respectively.

Recently, Huang and Liu [12] presented the following result, for , that

(1.7)

where

(1.8)

In 2007, Kolotilina [13] also obtained some interesting results for a subclass, referred to as PBDD(), of the class of nonsingular -matrices.

Inspired by these works, we will exhibit some new upper and lower bounds for determinants with principal diagonal dominant and general -matrices by using different methods, which improve on the above inequalities (1.4)–(1.7). Finally, these bounds are used to localize some numerical characters, for example, the minimum eigenvalues, singular values, the condition number of matrix, and so forth.

This paper is organized as follows. In Section 2, we present some notations and preliminary results for certain determinants by using different methods. Subsequently, we apply them to estimate for some bounds of some numerical characters of matrices in Section 3.

2. Estimations for Matrix Determinants

First, let us consider the problem on the signs of determinants.

Lemma 2.1 (see [14]).

Let and let be an -matrix; if , then and .

Theorem 2.2.

For any nonsingular -matrix, ; if its determinant is a real number, then () if and only if .

Proof.

For any , set , where

(2.1)

Since is a nonsingular -matrix and , then by Lemma 2.1, and nonsingular.

Note that is a continuous function in ; if

(2.2)

then there exists a real number such that , which is contrary to the fact that is nonsingular. Therefore, we have that

(2.3)

Thus, the proof is completed.

Remark 2.3.

For these results in this paper, one can obtain much sharper estimates on determinants by computing the signs of determinants using Theorem 2.2.

Next, we establish some new bounds of determinants by using different techniques.

2.1. Determinants and Inverses of Matrices

In this section, we firstly give some lemmas, involving about some inequalities for the entries of matrix . They will be useful in the following proofs.

In addition, for convenience, we will denote by the principal submatrix of formed from all rows and all columns with indices between and inclusively, for example, is the submatrix of obtained by deleting the first row and the first column of .

Lemma 2.4 (see [15]).

Let ; if , then exists and

(2.4)

where

(2.5)

Theorem 2.5.

Let ; if , then

(2.6)

Proof.

Our method is very simple. Note that

(2.7)

where denotes the submatrix of obtained by deleting row and column . So, we obtain by Lemma 2.4, for ,

(2.8)

Since , then . Thus if one applies the induction with respect to to , by using (2.8), then it is not difficult to get the left inequality of (2.6). Similarly, the right inequality of (2.6) can be also proved.

Note that, for each , the row dominance factor for does not exceed the corresponding factor for (assuming that the original row indices of remain "attached" to the rows in ). Hence the following corollary is obvious.

Corollary 2.6.

If , then

(2.9)

Remark 2.7.

Obviously, the above results improve the inequalities (1.4)–(1.7). In fact, if and is nonsingular, then (for any ). By continuity, one knows that Theorem 2.5 and Corollary 2.6 hold for any nonsingular , too.

In addition, if is a nonsingular -matrix, then we can obtain the sharper bounds.

Theorem 2.8.

If is a nonsingular -matrix, then

(2.10)

Especially, one has that

(2.11)

where is defined by the following recursive equations:

(2.12)

Proof.

First, by Theorem 2.2, one knows that . Second, by [16, Lemma ], we have that

(2.13)

Similar to the proof of Theorem 2.5, one may deduce inequality (2.10).

Finally, by the definition of , it is easy to see that

(2.14)

Therefore, inequality (2.11) is obvious. The proof is completed.

In addition, it is worthy to mention that there exist some other choices for the number in Lemma 2.4 and Theorem 2.5, which may be better than for some matrices. But they seem complicated for the computation. For example, the number in [15, 17] is given as

(2.15)

where

(2.16)

2.2. Determinants and the Max-Norm

Now let us consider relationships between determinants and the max-norm of matrices.

Here, for a vector and a matrix , and mean and , respectively.

Lemma 2.9.

Let be an matrix and , then

(2.17)

where .

Proof.

By the definition of the matrix norm, there exists an -dimensional vector with such that

(2.18)

where and . Now denote that , then

(2.19a)

(2.19b)

that is,

(2.20a)

(2.20b)

Then substituting (2.20a) in (2.20b), we have that

(2.21)

The proof is completed.

Corollary 2.10.

If () is an matrix, then

(2.22)

Remark 2.11.

In [18], Corollary 2.10 has been proved in the case. In fact, one can easily prove that the bound (2.22) is better than the following classical Ahlberg-Nilson-Varah [19] bound for any matrix:

(2.23)

The following theorem is analogous to (1.7).

Theorem 2.12.

Let be an matrix, then

(2.24)

where

(2.25)

Proof.

Let

(2.26)

where , . By Schur's Theorem (see [5]), we have that

(2.27)

Then

(2.28)

In addition, by Lemma 2.9, we get that

(2.29)

Thus substituting (2.29) in (2.28) and applying (2.28) to , one can get inequality (2.24) by induction. The proof is completed.

In fact, the problem of bounding satisfying certain assumptions was considered in some literature; see [13, 16, 18, 20]. Recently, Kolotilina [21] also obtained some interesting results for the so-called - and -matrices, which form a subclass of nonsingular - and -matrices, respectively.

Let , , and let

(2.30)

be a partitioning of the index set into disjoint nonempty subsets. Denote that and represent in the following block form:

(2.31)

Then the following result was obtained in [21].

Lemma 2.13 (see [21]).

For the block matrix defined by (2.31), let be nonsingular for all . Then

(2.32)

where

(2.33)

Obviously, by Lemma 2.13, many of results on can be generalized to the block case. Note that one usually needs to compute many good inverses of submatrices for a large matrix. However, the result (2.32) can not be improved to , since we have that , where

(2.34)

For example, let us consider the following block-partitioned matrix:

(2.35)

It is easy to compute that , but . In the same time, it shows that the bound (2.32) is sharp.

2.3. Determinants of - and -Matrices

First, according to the proof of Theorem 2.12, one may further obtain the following conclusion for general -matrices.

Theorem 2.14.

If is a nonsingular -matrix, then

(2.36)

where

(2.37)

Proof.

Let be partitioned into

(2.38)

then

(2.39)

Let , where , , and are diagonal, strict upper and strict lower triangular parts of , respectively. Since and is also nonsingular -matrix, then, by Lemma 2.1, . So

(2.40)

Denote that , then , that is,

(2.41)

So

(2.42)

thus,

(2.43)

Applying the induction with respect to , to , the result (2.36) is obtained and the proof is completed.

In the above proof, if we replace with , then the following result can be obtained.

Corollary 2.15.

If is a nonsingular -matrix, then

(2.44)

or

(2.45)

Corollary 2.16.

If is a nonsingular -matrix, then exists and

(2.46)

Next, let us consider some -matrices. For a general -matrix , as it is well known, there exists a positive diagonal matrix such that . For example, the following -strictly diagonally dominant matrices (-) are illustrated.

Definition 2.17 (see [6, 22]).

Given any nonempty subset of , let denote its complement in . If () satisfies

(1), for all ,

(2), for all ,

then is said to be an - matrix and nonsingular, where

(2.47)

For - matrices, we may let , where

(2.48)

(2.49)

then . In addition, analogous to [12], one may choose the permutation matrix such that

(2.50)

where the entries are monotonically ordered as . Thus, all results on also hold for the general -matrix , since . For example, we apply Corollary 2.6 to , then we obtain the following result.

Theorem 2.18.

For an -matrix , it holds that

(2.51)

where , , and ; is as in (2.50).

Finally, it is mentioned that, for many matrices which are not diagonally dominant, specially when the off-diagonal entries of each row have close values, one may make use of -matrices to obtain better lower bounds of determinant.

Definition 2.19 (see [23, 24]).

Let , then one may write it as , where

(2.52)

If , then is called a -matrix (denoted by ) and , where , .

For example, let us consider the following two matrices:

(2.53)

Obviously, they are not strictly diagonally dominant matrices neither are they -matrices. Now by Definition 2.19, we, respectively, have that

(2.54)

that is, and are both -matrices. According to Corollary 2.6 (or other results) and Definition 2.19, we further have that

(2.55)

3. Applications to Some Estimations for Numerical Characters of Matrices

In this section, we will apply some results in Section 2 to get some simple and interesting estimates for some numerical characters of matrices. Regarding other applications such as the stability of finite and infinite dimensional systems and the solutions of nonlinear equations of mathematical physics, we refer readers to [1, 2, 25] for full details.

For convenience, we will use the following notations and definitions. For , we denote by , and , for all , Frobenius norm, its eigenvalues and singular values, respectively. And we assume that

(3.1)

Set , for any , and define for all . Clearly, the value of or can serve as a kind of measure for the nonsingularity of . Especially, the smallest eigenvalue can characterize certain properties of corresponding physical systems. For example, it represents decay rates of signals in linear electrical circuits [26]. In this section, we find their lower bounds.

Theorem 3.1.

If , then exists and

(3.2)

where is as in Lemma 2.4. If , then

(3.3)

where , , and are as in Theorem 2.18.

Proof.

For (3.2), by [27, Theorem ], we have that

(3.4)

Note that . According to Lemma 2.4 or Theorem 2.5, the conclusion (3.2) can be easily followed.

Similarly, by [27, Theorem ],

(3.5)

and by Theorem 2.18, the result (3.3) also holds.

In addition, we may apply the above results to investigate the estimation of singular values and the condition number , since, for any matrix ,

(3.6)

(3.7)

(see [5]). For example, applying Corollary 2.6 to (3.6) and (3.7), respectively, we have the following results.

Theorem 3.2.

If , then

(3.8)

(3.9)

Finally, let us recall another result on the estimate of the smallest singular values. In 2002, an interesting result for a nonsingular complex matrix of order as a function of , , and singular values is due to Piazza and Politi [28]:

(3.10)

where .

So, by (3.10), let and note that (3.6), then

(3.11)

which implies that

(3.12)

Applying those inequalities on determinants in Section 2 to (3.12), one may further obtain many more interesting conclusions.