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On improvements of Fischer’s inequality and Hadamard’s inequality for -matrices
Journal of Inequalities and Applications volume 2013, Article number: 460 (2013)
In this paper, the class of -matrices, which includes positive definite matrices, totally positive matrices, M-matrices and inverse M-matrices, is first introduced and the refinements of Fischer’s inequality and Hadamard’s inequality for -matrices are obtained. Some previous well-known results for totally nonnegative matrices can be regarded as the special case of this paper.
All matrices considered in this paper are real. For an matrix , the submatrix of A lying in rows indexed by α and the columns indexed by β will be denoted by . If , then the principal submatrix is abbreviated to . For any , let denote the complement of α relative to , and let denote the cardinality of α. If , we define . We use for the symmetric group on .
An matrix A is called a -matrix (P-matrix) if all the principal minors of A are nonnegative (positive). A P-matrix A is called 1-minor symmetric if , whenever . Of course, each of the P-matrices, such as positive definite matrices (PD), totally positive matrices (TP), M-matrices (M) and inverse M-matrices (), is 1-minor symmetric, see . We have known that the following multiplicative principal minor inequalities are classical for PD, M and matrices:
Fischer: , for ;
Koteljanskii: , for .
The study of multiplicative principal minor inequalities has been actively going on for many years, many authors have done various wonderful works on this topic, see [7–11]. In , Zhang and Yang improved Hadamard’s inequality for totally nonnegative matrices as follows:
If is an totally nonnegative matrix with , then
Hadamard’s inequality for some subclasses of -matrices is an important inequality in matrix analysis, inequality (1.1) is the generalization of Hadamard’s inequality for totally nonnegative matrices. It is a noticeable problem to generalize inequality (1.1) for totally nonnegative matrices to other classes of matrices. In this paper we give some new upper bounds of Fischer’s inequality and Hadamard’s inequality for a subclass of -matrices and extend the corresponding results due to Zhang and Yang (see ).
2 Some lemmas
To avoid triviality, we always assume . We will need important Sylvester’s identity for determinants (see ).
Lemma 2.1 
Let A be an matrix, , and suppose (). Define the matrix , with , by setting for every . B is called the Sylvester matrix of A associated with . Then Sylvester’s identity states that for each , with ,
For convenience, we introduce the following definition.
Definition 2.1 A -matrix (P-matrix) A is called a -matrix (K-matrix) if every principal submatrix of A satisfies Koteljanskii’s inequality.
Obviously, each principal submatrix of a -matrix (K-matrix) is a -matrix (K-matrix). Of course, each of the matrices PD, TP, M and is a K-matrix, the totally nonnegative matrices are -matrices. In fact, an evident necessary and sufficient condition for a K-matrix was given in .
Lemma 2.2 
A P-matrix satisfies Koteljanskii’s inequality if and only if it is 1-minor symmetric.
There are K-matrices that lie in none of the classes PD, TP, M and , e.g.,
Lemma 2.3 If an matrix is a -matrix, then for .
Proof For , , we consider the submatrix of the -matrix A. Since is a -matrix, by the Hadamard’s inequality, we have
This completes the proof. □
Lemma 2.4 Let A be a K-matrix, B be the Sylvester matrix of A associated with , then B is a K-matrix.
Proof For , with , by Sylvester’s identity (2.1), we have
this means that B is a P-matrix.
For , , with , by Sylvester’s identity (2.1), we have
By Lemma 2.2 and (2.2), we conclude that B is 1-minor symmetric, therefore B is a K-matrix. □
Lemma 2.5 If is an -matrix, then
Proof For each , by Lemma 2.3, we obtain
Since each principal minor of A is nonnegative, we have
hence inequality (2.3) follows. □
3 Main results
In this section, we give some new upper bounds for Fischer’s inequality and Hadamard’s inequality, and extend the corresponding results due to Zhang and Yang (see ).
Theorem 3.1 If is an -matrix, with , then
Proof If A is singular, (3.1) is valid obviously. If A is a nonsingular -matrix, by Fisher’s inequality, we know that A is a P-matrix, therefore A is a K-matrix. Now we prove (3.1) for the K-matrix by induction on n. When , it is very easy to see that the result is valid. We suppose that the result is valid for all ( and ) K-matrices, let k be the cardinality of α. Note that α and are symmetric in inequality (3.1). Without loss of generality, we assume . Let
where and .
For any and , let , and be the Sylvester matrix of A associated with . By Lemma 2.4, S is a K-matrix. Clearly, , then, by Fisher’s inequality, we have
where . It follows from Sylvester’s identity (2.1) and (3.2) that
Let , from Sylvester’s identity (2.1), we obtain
Thus, by the inductive hypothesis, we have
From (3.3), (3.4) and (3.5), it follows that
Combining (3.6) and (3.7), we obtain inequality (3.1). □
Example 3.1 Now, we consider a K-matrix that lies in none of the classes PD, TP, M and . Let
Then A is 1-minor symmetric and A is a P-matrix. By Lemma 2.2, we know that A is a K-matrix. Let , then . By calculating, we have
therefore inequality (3.1) holds.
Let (), by Theorem 3.1 and the induction, we can obtain the following conclusion.
Corollary 3.2 If is an -matrix, with , then
If is a totally nonnegative matrix with , Corollary 3.2 is certainly valid, so Corollary 3.2 is the generalization of Theorem 3 in .
Theorem 3.3 If is an -matrix, then
Proof If A is singular, by Lemma 2.3 and Lemma 2.5, we know (3.8) is valid. If A is a nonsingular -matrix, then A is a K-matrix. We suppose that A is a K-matrix in the following.
If , then equality in (3.8) holds. If , then first we prove the following conclusion:
For any , there exists such that
Case 1. If . For any , by Lemma 2.3, we know that (3.9) certainly holds.
Case 2. If . Let , then , it is easy to see that (3.9) is true even with the equality sign for . Now we assume that .
Suppose that there exists , for any , such that
Then multiplying all the possible inequalities in (3.10) yields
Let . If , by (3.11) we have
If , by (3.11) we have
By (3.12) and (3.13), we have
On the other hand, since the principal submatrix of A is a K-matrix, applying Lemma 2.5 to yields
which is a contradiction to (3.14), therefore (3.9) holds.
By Fisher’s inequality (3.1) and (3.9), we have
therefore (3.8) holds. □
If is a totally nonnegative matrix, Theorem 3.3 is certainly valid too, so Theorem 3.3 is the generalization of Theorem 4 in .
Example 3.2 Now we consider the previous K-matrix. Let
By calculating we have
therefore inequality (3.8) holds.
By Corollary 3.2 and Theorem 3.3, we get the following result.
Corollary 3.4 If is an -matrix, with , then
If A is a totally nonnegative matrix, Corollary 3.4 is valid, so we obtain the result in .
Corollary 3.5 
If is an totally nonnegative matrix, with , then
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The author expresses his deep gratitude to the referees for their many very valuable suggestions and comments. The research of this paper was supported by the Natural Science Foundation of Shandong Province of China (ZR2010AL017).
The author declares that he has no competing interests.
The studies and manuscript of this paper was written by YL independently.
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Li, Y. On improvements of Fischer’s inequality and Hadamard’s inequality for -matrices. J Inequal Appl 2013, 460 (2013). https://doi.org/10.1186/1029-242X-2013-460
- totally nonnegative matrix
- Fischer’s inequality
- Hadamard’s inequality