Skip to content


  • Research Article
  • Open Access

On Several Matrix Kantorovich-Type Inequalities

Journal of Inequalities and Applications20102010:571629

  • Received: 5 September 2009
  • Accepted: 1 February 2010
  • Published:


We present several matrix Kantorovich-type inequalities, which improve the results obtained in Liu and Neudecker (1996). Elementary methods suffice to prove the inequalities.


  • General Version
  • Matrix Version
  • Kantorovich Inequality

1. Introduction

Let be a positive (semi-)definite Hermite matrix with eigenvalues contained in the interval where Let be matrix, and let denotes the column space of

A well-know matrix version of Kantorovich inequality asserts that (see[13])


for and where denotes the conjugate transpose of the matrix .

Let be an -by- matrix; the Moore-Penrose inverse of is defined as the unique -by- matrix satisfying all of the following four criteria (see, e.g., [4]):


It is not difficult to see that if , then we can get ; thus, for

In paper [5], from (which is equivalent to (13) in [6]), Liu and Neudecker presented the following so-called Kantorovich-type inequality:


for and and the following inequality:


for and Furthermore, in the same way, they obtained three more general versions.


for and

In the next section, we shall present several similar matrix Kantorovich-type inequalities, which improve some results above.

2. New Matrix Kantorovich-Type Inequalities

We first introduce two lemmas.

Lemma 2.1.

for and


It is easy to see that if then thus, we have


In [7], Dragomir defines a transform for this transform, we have the following lemma.

Lemma 2.2.

Let then

for and



From Lemma 2.2, we can easily get the inequality (1.4).

Corollary 3.2.

for and


From , we have

The proof is completed.

Theorem 2.4.

for and


From Lemmas 2.1 and 2.2, we have

The proof of Theorem 2.4 is completed.

Remark 2.5.

It is not difficult to see that if then we conclude that Theorem 2.4 gives an improvement of the Kantorovich inequality (1.3).

Furthermore, in similar way we got Theorem 2.4, and we obtain three more general versions, which also improve the inequalities (1.5), (1.6), (1.7), respectively.

Theorem 2.6.


for and where


In fact, they are equivalent by noting and For (2.9), pre- and postmultiplying by and respectively, we get the inequality (2.10); similarly, for (2.10), pre- and postmultiplying by respectively, we get the inequality (2.11). So, we only prove the inequality (2.9).

Similarly, with Lemma 2.2, we have


From the proof, it is easy to see that so, we conclude that the inequality (2.9) gives an improvement of the inequality (1.5), meanwhile, the inequalities (2.10) and (2.11) improve the inequalities (1.6) and (1.7), respectively.



The authors would like to thank the anonymous referees for their valuable comments which have been implemented in this revised version. This work is supported by Natural Science Foundation of China no.10961010, Natural Science Foundation of Jiangxi, China no 2007GZS1760, and scientific and technological project of Jiangxi education office, China no GJJ08432.

Authors’ Affiliations

School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
Department of Mathematics, Jiujiang University, Jiujiang, 332005, China
School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, 550001, China


  1. Mond B, Pečarić JE: A matrix version of the Ky Fan generalization of the Kantorovich inequality. Linear and Multilinear Algebra 1994, 36(3):217–221. 10.1080/03081089408818291MathSciNetView ArticleMATHGoogle Scholar
  2. Wang SG, Shao J: Constrained Kantorovich inequalities and relative efficiency of least squares. Journal of Multivariate Analysis 1992, 42(2):284–298. 10.1016/0047-259X(92)90048-KMathSciNetView ArticleMATHGoogle Scholar
  3. Chen L, Zeng X-M: Rate of convergence of a new type kantorovich variant of bleimann-butzer-hahn operators. Journal of Inequalities and Applications 2009, 2009:-10.Google Scholar
  4. Golub GH, Van Loan CF: Matrix Computation. Johns Hopkins University, Baltimore, Md, USA; 1983.MATHGoogle Scholar
  5. Liu S, Neudecker H: Several matrix Kantorovich-type inequalities. Journal of Mathematical Analysis and Applications 1996, 197(1):23–26. 10.1006/jmaa.1996.0003MathSciNetView ArticleMATHGoogle Scholar
  6. Marshall AW, Olkin I: Matrix versions of the Cauchy and Kantorovich inequalities. Aequationes Mathematicae 1990, 40(1):89–93. 10.1007/BF02112284MathSciNetView ArticleMATHGoogle Scholar
  7. Dragomir SS: New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. Linear Algebra and Its Applications 2008, 428(11–12):2750–2760. 10.1016/j.laa.2007.12.025MathSciNetView ArticleMATHGoogle Scholar


© Zhibing Liu et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.