- Open Access
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.
- Hilbert Space
- Matrix Version
- Hermitian Matrix
- Nonzero Vector
- Complex Hilbert Space
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).
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).
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