Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 571629, 5 pages
doi:10.1155/2010/571629
Research Article

On Several Matrix Kantorovich-Type Inequalities

Zhibing Liu,1,2 Linzhang Lu,1,3 and Kanmin Wang2

1School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2Department of Mathematics, Jiujiang University, Jiujiang 332005, China
3School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China

Received 5 September 2009; Revised 5 January 2010; Accepted 1 February 2010

Academic Editor: Stevo Stević

Copyright © 2010 Zhibing Liu et al. 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.

Abstract

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

1. Introduction

Let 𝐴 ∈ 𝑀 𝑛 be a positive (semi-)definite Hermite matrix with eigenvalues contained in the interval [ 𝑚 , 𝑀 ] , where 0 < 𝑚 < 𝑀 . Let 𝑉 be 𝑛 × 𝑟 matrix, and let ℜ ( 𝐴 ) denotes the column space of 𝐴 .

A well-know matrix version of Kantorovich inequality asserts that (see[1–3])

𝑉 ∗ 𝐴 2 𝑉 ≤ ( 𝑚 + 𝑀 ) 2 4 𝑚 𝑀 ( 𝑉 ∗ 𝐴 𝑉 ) 2 , ( 1 . 1 ) for 𝐴 > 0 and 𝑉 ∗ 𝑉 = 𝐼 , where 𝑉 ∗ denotes the conjugate transpose of the matrix 𝑉 .

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

𝐵 𝐵 + 𝐵 = 𝐵 , 𝐵 + 𝐵 𝐵 + = 𝐵 + , ( 𝐵 𝐵 + ) ∗ = 𝐵 𝐵 + , ( 𝐵 + 𝐵 ) ∗ = 𝐵 + 𝐵 . ( 1 . 2 )

It is not difficult to see that if 𝑉 ∗ 𝑉 = 𝐼 , then 𝑉 𝑉 ∗ = 𝑉 𝑉 + ≤ 𝐼 ; we can get 𝑉 ∗ 𝐴 𝐴 𝑉 ≥ 𝑉 ∗ 𝐴 𝑉 𝑉 ∗ 𝐴 𝑉 ; thus, 𝑉 ∗ 𝐴 2 𝑉 − ( 𝑉 ∗ 𝐴 𝑉 ) 2 ≥ 0 , for 𝐴 > 0 .

In paper [5], from 𝐴 2 ≤ ( 𝑚 + 𝑀 ) 𝐴 − 𝑚 𝑀 𝐼 (which is equivalent to (13) in [6]), Liu and Neudecker presented the following so-called Kantorovich-type inequality:

𝑉 ∗ 𝐴 2 𝑉 − ( 𝑉 ∗ 𝐴 𝑉 ) 2 ≤ ( 𝑀 − 𝑚 ) 2 4 𝐼 ( 1 . 3 ) for 𝐴 > 0 and 𝑉 ∗ 𝑉 = 𝐼 , and the following inequality:

( 𝑉 ∗ 𝐴 2 𝑉 ) 1 / 2 ≤ ( 𝑚 + 𝑀 ) 2 √ 𝑉 𝑚 𝑀 ∗ 𝐴 𝑉 ( 1 . 4 ) for 𝐴 > 0 and 𝑉 ∗ 𝑉 = 𝐼 . Furthermore, in the same way, they obtained three more general versions.

𝑉 𝑉 + 𝐴 2 𝑉 𝑉 + − ( 𝑉 𝑉 + 𝐴 𝑉 𝑉 + ) 2 ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + , 𝑉 ( 1 . 5 ) ∗ 𝐴 2 𝑉 − 𝑉 ∗ 𝐴 𝑉 𝑉 + 1 𝐴 𝑉 ≤ 4 ( 𝑀 − 𝑚 ) 2 𝑉 ∗ 𝑉 𝑉 , ( 1 . 6 ) + 𝐴 2 𝑉 + ∗ − 𝑉 + 𝐴 𝑉 𝑉 + 𝐴 𝑉 + ∗ ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + ( 1 . 7 ) for 𝐴 > 0 and 𝑉 ∈ ℜ ( 𝐴 ) .

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. 0 ≤ ( 𝑀 𝐼 − 𝑉 ∗ 𝐴 𝑉 ) ( 𝑉 ∗ 𝐴 𝑉 − 𝑚 𝐼 ) ≤ ( 1 / 4 ) ( 𝑀 − 𝑚 ) 2 𝐼 , for 𝐴 > 0 and 𝑉 ∗ 𝑉 = 𝐼 .

Proof. It is easy to see that if 𝑚 𝐼 ≤ 𝐴 ≤ 𝑀 𝐼 , then 𝑚 𝐼 ≤ 𝑉 ∗ 𝐴 𝑉 ≤ 𝑀 𝐼 ; thus, we have 0 ≤ 𝑀 𝐼 − 𝑉 ∗ 𝑉 𝐴 𝑉 ∗ 𝐴 𝑉 − 𝑚 𝐼 = ( 𝑚 + 𝑀 ) 𝑉 ∗ 𝐴 𝑉 − 𝑚 𝑀 𝐼 − ( 𝑉 ∗ 𝐴 𝑉 ) 2 = 1 4 ( 𝑀 − 𝑚 ) 2 𝑉 𝐼 − ∗ 1 𝐴 𝑉 − 2 ( 𝑚 + 𝑀 ) 𝐼 2 ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝐼 , ( 2 . 1 ) for 𝑉 ∗ 𝑉 = 𝐼 .

In [7], Dragomir defines a transform 𝐶 𝑚 , 𝑀 ( 𝐴 ) = ( 𝐴 − 𝑚 𝐼 ) ( 𝑀 𝐼 − 𝐴 ) ; for this transform, we have the following lemma.

Lemma 2.2. Let 𝐶 ( 𝐴 , 𝑉 ) = 𝑉 ∗ ( 𝐴 − 𝑚 𝐼 ) ( 𝑀 𝐼 − 𝐴 ) 𝑉 ; then 1 𝐶 ( 𝐴 , 𝑉 ) = 4 ( 𝑀 − 𝑚 ) 2 𝐼 − 𝑉 ∗ 1 𝐴 − 2 ( 𝑚 + 𝑀 ) 𝐼 2 𝑉 ; ( 2 . 2 ) thus 1 0 ≤ 𝐶 ( 𝐴 , 𝑉 ) ≤ 4 ( 𝑀 − 𝑚 ) 2 𝐼 ( 2 . 3 ) for 𝐴 > 0 and 𝑉 ∗ 𝑉 = 𝐼 .

Proof. 𝐶 ( 𝐴 , 𝑉 ) = 𝑉 ∗ ( 𝐴 − 𝑚 𝐼 ) ( 𝑀 𝐼 − 𝐴 ) 𝑉 = 𝑉 ∗ 𝑀 − 𝑚 2 𝐼 + 𝐴 − 𝑀 + 𝑚 2 𝐼 𝑀 − 𝑚 2 𝐼 − 𝐴 − 𝑀 + 𝑚 2 𝐼 𝑉 = 1 4 ( 𝑀 − 𝑚 ) 2 𝐼 − 𝑉 ∗ 1 𝐴 − 2 ( 𝑚 + 𝑀 ) 𝐼 2 𝑉 ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝐼 , f o r 𝑉 ∗ 𝑉 = 𝐼 . ( 2 . 4 )

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

Corollary 3.2. ( 𝑉 ∗ 𝐴 2 𝑉 ) 1 / 2 √ ≤ ( 𝑚 + 𝑀 ) / ( 2 𝑚 𝑀 ) 𝑉 ∗ 𝐴 𝑉 , for 𝐴 > 0 and 𝑉 ∗ 𝑉 = 𝐼 .

Proof. From 𝐶 ( 𝐴 , 𝑉 ) ≥ 0 , we have ( 𝑚 + 𝑀 ) 𝑉 ∗ 𝐴 𝑉 − 𝑉 ∗ 𝐴 2 𝑉 − 𝑚 𝑀 𝐼 ≥ 0 ; ( 2 . 5 ) then ( 𝑚 + 𝑀 ) 𝑉 ∗ 𝐴 𝑉 ≥ 𝑉 ∗ 𝐴 2 √ 𝑉 + 𝑚 𝑀 𝐼 ≥ 2 𝑚 𝑀 ( 𝑉 ∗ 𝐴 2 𝑉 ) 1 / 2 . ( 2 . 6 ) The proof is completed.

Theorem 2.4. 𝑉 ∗ 𝐴 2 𝑉 − ( 𝑉 ∗ 𝐴 𝑉 ) 2 ≤ ( 1 / 4 ) ( 𝑀 − 𝑚 ) 2 𝐼 − 𝐶 ( 𝐴 , 𝑉 ) for 𝐴 > 0 and 𝑉 ∗ 𝑉 = 𝐼 .

Proof. 𝑉 ∗ 𝐴 2 𝑉 − ( 𝑉 ∗ 𝐴 𝑉 ) 2 = 𝑉 ∗ 𝐴 2 𝑉 + 𝑚 𝑀 𝐼 − ( 𝑚 + 𝑀 ) 𝑉 ∗ 𝑉 𝐴 𝑉 − ∗ 𝐴 𝑉 2 + 𝑚 𝑀 𝐼 − ( 𝑚 + 𝑀 ) 𝑉 ∗ = 𝐴 𝑉 𝑀 𝐼 − 𝑉 ∗ 𝑉 𝐴 𝑉 ∗ 𝐴 𝑉 − 𝑚 𝐼 − 𝑉 ∗ ( 𝐴 − 𝑚 𝐼 ) ( 𝑀 𝐼 − 𝐴 ) 𝑉 . ( 2 . 7 ) From Lemmas 2.1 and 2.2, we have 𝑉 ∗ 𝐴 2 𝑉 − ( 𝑉 ∗ 𝐴 𝑉 ) 2 ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝐼 − 𝐶 ( 𝐴 , 𝑉 ) . ( 2 . 8 ) The proof of Theorem 2.4 is completed.

Remark 2.5. It is not difficult to see that if 𝑉 ∗ 𝐴 2 𝑉 − ( 𝑉 ∗ 𝐴 𝑉 ) 2 ≤ ( 1 / 4 ) ( 𝑀 − 𝑚 ) 2 𝐼 − 𝐶 ( 𝐴 , 𝑉 ) ≤ ( 1 / 4 ) ( 𝑀 − 𝑚 ) 2 𝐼 , 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. 𝑉 𝑉 + 𝐴 2 𝑉 𝑉 + − ( 𝑉 𝑉 + 𝐴 𝑉 𝑉 + ) 2 ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + − 𝐶 𝐴 , 𝑉 , 𝑉 + , 𝑉 ( 2 . 9 ) ∗ 𝐴 2 𝑉 − 𝑉 ∗ 𝐴 𝑉 𝑉 + 1 𝐴 𝑉 ≤ 4 ( 𝑀 − 𝑚 ) 2 𝑉 ∗ 𝑉 − 𝐶 𝐴 , 𝑉 ∗ , 𝑉 , 𝑉 ( 2 . 1 0 ) + 𝐴 2 𝑉 + ∗ − 𝑉 + 𝐴 𝑉 𝑉 + 𝐴 𝑉 + ∗ ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + ∗ − 𝐶 𝐴 , 𝑉 , 𝑉 + ∗ ( 2 . 1 1 ) for 𝐴 > 0 and 𝑉 ∈ ℜ ( 𝐴 ) , where 𝐶 ( 𝐴 , 𝑉 , 𝑈 ) = 𝑉 𝑈 ( 𝐴 − 𝑚 𝐼 ) ( 𝑀 𝐼 − 𝐴 ) 𝑉 𝑈 , 𝑈 ∈ 𝐶 𝑟 × 𝑛 .

Proof. In fact, they are equivalent by noting 𝑉 ∗ = 𝑉 ∗ 𝑉 𝑉 + and 𝑉 + = 𝑉 + 𝑉 + ∗ 𝑉 ∗ . For (2.9), pre- and postmultiplying by 𝑉 ∗ and 𝑉 , respectively, we get the inequality (2.10); similarly, for (2.10), pre- and postmultiplying by 𝑉 + 𝑉 + ∗ , respectively, we get the inequality (2.11). So, we only prove the inequality (2.9).
Similarly, with Lemma 2.2, we have 0 ≤ 𝐶 𝐴 , 𝑉 , 𝑉 + = 1 4 ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + − 𝑉 𝑉 + 1 𝐴 − 2 ( 𝑚 + 𝑀 ) 𝐼 2 𝑉 𝑉 + ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + , 𝑉 𝑉 + 𝐴 2 𝑉 𝑉 + − ( 𝑉 𝑉 + 𝐴 𝑉 𝑉 + ) 2 = 𝑉 𝑉 + 𝐴 2 𝑉 𝑉 + + 𝑚 𝑀 𝑉 𝑉 + − ( 𝑚 + 𝑀 ) 𝑉 𝑉 + 𝐴 𝑉 𝑉 + − 𝑉 𝑉 + 𝐴 𝑉 𝑉 + 2 + 𝑚 𝑀 𝑉 𝑉 + − ( 𝑚 + 𝑀 ) 𝑉 𝑉 + 𝐴 𝑉 𝑉 + = 𝑀 𝑉 𝑉 + − 𝑉 𝑉 + 𝐴 𝑉 𝑉 + 𝑉 𝑉 + 𝐴 𝑉 𝑉 + − 𝑚 𝑉 𝑉 + − 𝑉 𝑉 + ( 𝐴 − 𝑚 𝐼 ) ( 𝑀 𝐼 − 𝐴 ) 𝑉 𝑉 + , ≤ 1 4 ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + − 𝐶 𝐴 , 𝑉 , 𝑉 + . ( 2 . 1 2 )

Remark. From the proof, it is easy to see that 𝑉 𝑉 + 𝐴 2 𝑉 𝑉 + − ( 𝑉 𝑉 + 𝐴 𝑉 𝑉 + ) 2 ≤ ( 1 / 4 ) ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + − 𝐶 ( 𝐴 , 𝑉 , 𝑉 + ) ≤ ( 1 / 4 ) ( 𝑀 − 𝑚 ) 2 𝑉 𝑉 + ; 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.

Acknowledgments

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.

References

  1. B. Mond and J. E. Pečarić, “A matrix version of the Ky Fan generalization of the Kantorovich inequality,” Linear and Multilinear Algebra, vol. 36, no. 3, pp. 217–221, 1994.
  2. S. G. Wang and J. Shao, “Constrained Kantorovich inequalities and relative efficiency of least squares,” Journal of Multivariate Analysis, vol. 42, no. 2, pp. 284–298, 1992.
  3. L. Chen and X.-M. Zeng, “Rate of convergence of a new type kantorovich variant of bleimann-butzer-hahn operators,” Journal of Inequalities and Applications, vol. 2009, Article ID 852897, 10 pages, 2009.
  4. G. H. Golub and C. F. Van Loan, Matrix Computation, Johns Hopkins University, Baltimore, Md, USA, 1983.
  5. S. Liu and H. Neudecker, “Several matrix Kantorovich-type inequalities,” Journal of Mathematical Analysis and Applications, vol. 197, no. 1, pp. 23–26, 1996.
  6. A. W. Marshall and I. Olkin, “Matrix versions of the Cauchy and Kantorovich inequalities,” Aequationes Mathematicae, vol. 40, no. 1, pp. 89–93, 1990.
  7. S. S. Dragomir, “New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces,” Linear Algebra and Its Applications, vol. 428, no. 11-12, pp. 2750–2760, 2008.