- Research Article
- Open Access
On Several Matrix Kantorovich-Type Inequalities
© Zhibing Liu et al. 2010
- Received: 5 September 2009
- Accepted: 1 February 2010
- Published: 14 February 2010
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
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
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., ):
It is not difficult to see that if , then we can get ; thus, for
for and and the following inequality:
for and Furthermore, in the same way, they obtained three more general versions.
In the next section, we shall present several similar matrix Kantorovich-type inequalities, which improve some results above.
We first introduce two lemmas.
In , Dragomir defines a transform for this transform, we have the following lemma.
From Lemma 2.2, we can easily get the inequality (1.4).
The proof is completed.
The proof of Theorem 2.4 is completed.
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.
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).
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.
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