- Research Article
- Open Access
Some New Results on Determinantal Inequalities and Applications
© Hou-Biao Li et al. 2010
- Received: 23 August 2009
- Accepted: 28 December 2009
- Published: 12 January 2010
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.
- Mathematical Physic
- Nonempty Subset
- Small Eigenvalue
- Block Matrix
- Matrix Norm
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 . So it has hitherto great influence on every branch of mathematics (see, e.g., [1–4]).
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  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
In addition, there are various generalizations of class. Recall that a doubly strictly diagonally dominant( )  is a matrix such that for all , one has
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  proved, under the hypothesis , that
Subsequently, Price  suggested another new expression as
In , the above inequalities (1.4)-(1.5) are improved in such a way, for an arbitrary index , that
Recently, Huang and Liu  presented the following result, for , that
In 2007, Kolotilina  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.
First, let us consider the problem on the signs of determinants.
Lemma 2.1 (see ).
Thus, the proof is completed.
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 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 ).
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.
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.
Similar to the proof of Theorem 2.5, one may deduce inequality (2.10).
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
2.2. Determinants and the Max-Norm
Now let us consider relationships between determinants and the max-norm of matrices.
The proof is completed.
The following theorem is analogous to (1.7).
In fact, the problem of bounding satisfying certain assumptions was considered in some literature; see [13, 16, 18, 20]. Recently, Kolotilina  also obtained some interesting results for the so-called - and -matrices, which form a subclass of nonsingular - and -matrices, respectively.
Then the following result was obtained in .
Lemma 2.13 (see ).
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
For example, let us consider the following block-partitioned matrix:
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.
then . In addition, analogous to , one may choose the permutation matrix such that
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.
For example, let us consider the following two 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.
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 . In this section, we find their lower bounds.
Similarly, by [27, Theorem ],
and by Theorem 2.18, the result (3.3) also holds.
(see ). For example, applying Corollary 2.6 to (3.6) and (3.7), respectively, we have the following results.
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 :
which implies that
Applying those inequalities on determinants in Section 2 to (3.12), one may further obtain many more interesting conclusions.
The authors sincerely thank reviewers for valuable comments and suggestions, which led to a substantial improvement on the presentation of this paper. In addition, the authors are supported in part by NSFC (160973015), Sichuan Province Project for Applied Basic Research (2008JY0052), the Project for Academic Leader, Group of UESTC and Young Scholar Research Foundation of UESTC (L08011001JX0776), and the China Postdoctoral Science Foundation (20090460244).
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