- Open Access
On improvements of Fischer’s inequality and Hadamard’s inequality for -matrices
© Li; licensee Springer. 2013
- Received: 16 May 2013
- Accepted: 9 September 2013
- Published: 7 November 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.
- totally nonnegative matrix
- Fischer’s inequality
- Hadamard’s inequality
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 .
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:
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 ).
To avoid triviality, we always assume . We will need important Sylvester’s identity for determinants (see ).
Lemma 2.1 
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.
Lemma 2.3 If an matrix is a -matrix, then for .
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.
this means that B is a P-matrix.
By Lemma 2.2 and (2.2), we conclude that B is 1-minor symmetric, therefore B is a K-matrix. □
hence inequality (2.3) follows. □
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 ).
where and .
Combining (3.6) and (3.7), we obtain inequality (3.1). □
therefore inequality (3.1) holds.
Let (), by Theorem 3.1 and the induction, we can obtain the following conclusion.
If is a totally nonnegative matrix with , Corollary 3.2 is certainly valid, so Corollary 3.2 is the generalization of Theorem 3 in .
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:
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 .
which is a contradiction to (3.14), therefore (3.9) holds.
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 .
therefore inequality (3.8) holds.
By Corollary 3.2 and Theorem 3.3, we get the following result.
If A is a totally nonnegative matrix, Corollary 3.4 is valid, so we obtain the result in .
Corollary 3.5 
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).
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