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Notes on Greub-Rheinboldt inequalities
Journal of Inequalities and Applications volume 2013, Article number: 7 (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 .
An equivalent form of this result is the inequality
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]).
In 2008, Dragomir gave a refinement of the additive version of the operator Kantorovich inequality ,
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.
We adopt the following transform for a positive definite Hermitian matrix with eigenvalues :
Then the following inequality holds :
The result above is an improvement of the Kantorovich inequality (1.1).
A generalized form of the Kantorovich inequality presented by Greub and Rheinboldt  in 1959 is known as the Greub-Rheinboldt inequality in operator theoretic terms, which is also an important and early example of the so-called complementary inequality referred to in ,
where A and B are commuting positive definite self-adjoint operators on a Hilbert space, with upper and lower bounds and , , respectively.
In 1997, Fujii et al.  generalized the Greub-Rheinboldt inequality to pairs of invertible operators that may not even commute,
where A, B are invertible positive operators satisfying and , and . By using the viewpoint of interaction antieigenvalue, Gustafson  sharpened the Greub-Rheinboldt inequality (1.7) to obtain the following result:
where A and B are commuting positive definite self-adjoint operators on a Hilbert space.
Let A and B be two positive definite Hermite matrices and with real eigenvalues and , respectively. Moreover, let . Then a matrix version of (1.9) is
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.
Let V be an matrix such that , i.e., V is suborthogonal. Another well-known matrix version of the Kantorovich inequality asserts that
for any , , and .
Mond and Pečarić proved the following matrix version inequality (see (7) in ):
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)
Let be a positive definite Hermitian matrix. The following inequalities hold:
for any .
Let A, B be two invertible commuting Hermite matrices. Denote by and the eigenvalues of A and B, respectively. Then there exists a unitary matrix such that , , where , . Note that is a permutation of . Let (), then it is easy to see that all eigenvalues of are . Without loss of generality, we may assume that , and . For convenience, we introduce the notation
If , then we can define
Lemma 2.2 Let A and B be two positive definite commuting matrices with eigenvalues , , respectively. , and are as before. Then for any ,
for any .
Proof From (2.2),
Let . Thus, . Then
On the other hand,
The proof of is similar. □
Theorem 2.3 With the assumptions of Lemma 2.2,
Proof Let , . Then
From (1.2) and (1.6),
From (2.5) and (2.10), we have
By substituting (2.12) and (2.10) into (2.11), the inequality becomes
Corollary 2.4 Let A and B be two positive definite commuting matrices with eigenvalues , , respectively. Then
holds for any nonzero vector .
By Theorem 2.3, we have the following:
Let . It can be easily deduced that is monotone increasing on . Let , . From the definition of and , we know that . Thus,
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.
As is declared in (1.11), for , , and m, M mentioned above, the following inequality holds:
It is not difficult to see that as , then , where + indicates the Moore-Penrose inverse. Multiplying from the right and from the left by and AV respectively, we have for . From the well-known Löwner-Heinz inequality, we have and the following inequality (see in ):
For , , the convexity of implies that
If A has the representation , where Γ is unitary and , and if , , then from (2.16) it follows that
After multiplying from the right and from the left by Γ and , it is not difficult to see that (2.17) yields the following :
Based on (2.18), we derive several results on the inequality (1.12).
Theorem 2.5 For any and ,
Proof From (2.18) and , we can get
Since , (2.20) can be turned into
By adding to both sides of the inequality (2.21), we obtain that
Thus, we finally have
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.
Corollary 2.6 Let and be positive definite matrices with eigenvalues of contained in the interval . Then
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.
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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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Zhao, D., Li, H. & Gong, Z. Notes on Greub-Rheinboldt inequalities. J Inequal Appl 2013, 7 (2013). https://doi.org/10.1186/1029-242X-2013-7
- Hilbert Space
- Matrix Version
- Hermitian Matrix
- Nonzero Vector
- Complex Hilbert Space