Notes on Greub-Rheinboldt inequalities
© Zhao et al.; licensee Springer 2013
Received: 9 January 2012
Accepted: 10 December 2012
Published: 4 January 2013
In this paper, we focus on matrix Greub-Rheinboldt inequalities for commutative positive definite Hermitian matrix pairs. Some improvements, which yield sharpened bounds compared with existing results, are presented.
1 Introduction and preliminaries
Let denote the space of complex matrices and write . The identity matrix in is denoted by . As usual, denotes the conjugate transpose of the matrix A. A matrix is an Hermite matrix if . An Hermitian matrix A is said to be positive semi-definite or nonnegative definite, written as , if , . A is further called positive definite, symbolized , if for all nonzero . An equivalent condition for to be positive definite is that A is an Hermitian matrix and all eigenvalues of A are positive.
for any nonzero vector .
valid for any nonzero vector .
This famous inequality plays an important role in statistics (see [3, 4]; for the latest work on applications in statistics, we refer to Seddighin’s work ) and numerical analysis, for example, studying the rates of convergence and error bounds of solving systems of equations (see in [5, 6]).
where A is a self-adjoint bounded linear operator on a complex Hilbert space, , such that in the partial operator order, , and .
A further improvement of the matrix version of (1.3) is proposed in , where the classical Kantorovich inequality (1.1) is modified to apply not only to positive definite, but also to all invertible Hermitian matrices.
The result above is an improvement of the Kantorovich inequality (1.1).
where A and B are commuting positive definite self-adjoint operators on a Hilbert space, with upper and lower bounds and , , respectively.
where A and B are commuting positive definite self-adjoint operators on a Hilbert space.
for any nonzero vector .
In 2005, Seddighin  extended the Greub-Rheinboldt inequality (1.9) to pairs of normal operators and established for what vectors the Greub-Rheinboldt inequality becomes equality.
for any , , and .
In the next section, we propose some refinements about the matrix Kantorovich-type inequalities (1.2), the Greub-Rheinboldt inequality for commutative positive definite Hermitian matrix pairs, and (1.10) for positive definite matrices, yielding sharpened upper bounds compared with original results, together with an improvement to (1.12).
2 Main results
In this section, we first introduce some lemmas.
Lemma 2.1 (in , Lemma 2.2)
for any .
for any .
The proof of is similar. □
holds for any nonzero vector .
Remark From Lemma 2.2 and (2.15), we can obtain a sharpened bound for the classical Kantorovich-type inequality, i.e., the Greub-Rheinboldt inequality.
Besides the discussion on the Greub-Rheinboldt inequality (1.9), we are also interested in another form of Kantorovich-type inequality aforementioned. We turn our attention to the inequalities (1.11) and (1.12) in the remainder of this paper.
Let A be an positive (semi-) definite Hermitian matrix with (nonzero) eigenvalues contained in the interval , where . Let V be matrices.
Based on (2.18), we derive several results on the inequality (1.12).
where . □
Remark It is obvious that . Thus, Theorem 2.5 indeed presents an improvement of the Kantorovich-type inequality (1.12) in .
For an application to the Hadamard product, we have the following corollary.
where V is the selection matrix of order with the property (⊗ and ∘ indicate the tensor and the Hadamard product, respectively).
In this paper, we introduce some new bounds for several Kantorovich-type inequalities for commutative positive definite Hermitian matrix pairs. As a particular situation, in Corollary 2.4, when A and B are both positive definite, the result provides a sharpened upper bound for the matrix version of the well-known Greub-Rheinboldt inequality. Moreover, it holds for negative definite Hermite matrices. Also, a refinement of Kantorovich-type inequalities concerning positive definite matrices is presented together with an application to the Hadamard product.
The authors would like to thank all the reviewers who read this paper carefully and provided valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (Grant No. 60831001).
- Greub W, Rheinboldt W: On a generalization of an inequality of L.V. Kantorovich. Proc. Am. Math. Soc. 1959, 10: 407–415. 10.1090/S0002-9939-1959-0105028-3MATHMathSciNetView ArticleGoogle Scholar
- Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985.MATHView ArticleGoogle Scholar
- Seddighin M: Antieigenvalue techniques in statistics. Linear Algebra Appl. 2009, 430(10):2566–2580. 10.1016/j.laa.2008.05.007MATHMathSciNetView ArticleGoogle Scholar
- Wang S-G: A matrix version of the Wielandt inequality and its applications to statistics. Linear Algebra Appl. 1999, 296(1–3):171–181. 10.1016/S0024-3795(99)00117-2MATHMathSciNetView ArticleGoogle Scholar
- Householder AS: The Theory of Matrices in Numerical Analysis. Blaisdell, New York; 1964.MATHGoogle Scholar
- Nocedal J: Theory of algorithms for unconstrained optimization. Acta Numer. 1992, 1: 199–242.MathSciNetView ArticleGoogle Scholar
- Dragomir SS: New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. Linear Algebra Appl. 2008, 428(11–12):2750–2760. 10.1016/j.laa.2007.12.025MATHMathSciNetView ArticleGoogle Scholar
- Liu Z, Wang K, Xu C: A note on Kantorovich inequality for Hermite matrices. J. Inequal. Appl. 2011., 2011: Article ID 245767Google Scholar
- Gustafson K: Interaction antieigenvalues. J. Math. Anal. Appl. 2004, 299(1):174–185. 10.1016/j.jmaa.2004.06.012MATHMathSciNetView ArticleGoogle Scholar
- Fujii M, Izumino S, Nakamoto R, Seo Y: Operator inequalities related to Cauchy-Schwarz and Holder-McCarthy inequalities. Nihonkai Math. J. 1997, 8(2):117–122.MATHMathSciNetGoogle Scholar
- Seddighin M: On the joint antieigenvalue of operators on normal subalgebras. J. Math. Anal. Appl. 2005, 312(1):61–71. 10.1016/j.jmaa.2005.03.021MATHMathSciNetView ArticleGoogle Scholar
- Mond B, Peǎarić JE: A matrix version of the Ky Fan generalization of the Kantorovich inequality. Linear Multilinear Algebra 1994, 36(3):217–221. 10.1080/03081089408818291MATHView ArticleGoogle Scholar
- Wang SG, Shao J: Constrained Kantorovich inequalities and relative efficiency of least squares. J. Multivar. Anal. 1992, 42(2):284–298. 10.1016/0047-259X(92)90048-KMATHMathSciNetView ArticleGoogle Scholar
- Chen L, Zeng XM: Rate of convergence of a new type Kantorovich variant of Bleimann-Butzerhahn operators. J. Inequal. Appl. 2009., 2009: Article ID 852897Google Scholar
- Yuan JT, Gao ZS: Complete form of Furuta inequality. Proc. Am. Math. Soc. 2008, 136(8):2859–2867. 10.1090/S0002-9939-08-09446-XMATHMathSciNetView ArticleGoogle Scholar
- Liu S, Neudecker H: Several matrix Kantorovich-type inequalities. Aequ. Math. 1990, 40(1):89–93. 10.1007/BF02112284MathSciNetView ArticleGoogle Scholar
- Marshall AW, Olkin I: Matrix versions of the Cauchy and Kantorovich inequalities. J. Math. Anal. Appl. 1996, 197(1):23–26. 10.1006/jmaa.1996.0003MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.