Notes on Greub-Rheinboldt inequalities

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.


Introduction and preliminaries
Let M m,n denote the space of m × n complex matrices and write M n ≡ M n,n . The identity matrix in M n is denoted by I n . As usual, A * = (Ā) T denotes the conjugate transpose of the matrix A. A matrix A ∈ M n is an Hermite matrix if A * = A. An Hermitian matrix A is said to be positive semi-definite or nonnegative definite, written as A ≥ , if x * Ax ≥ , ∀x ∈ C n . A is further called positive definite, symbolized A > , if x * Ax >  for all nonzero x ∈ C n . An equivalent condition for A ∈ M n to be positive definite is that A is an Hermitian matrix and all eigenvalues of A are positive.
Denote by λ  ≤ λ  ≤ · · · ≤ λ n the eigenvalues of an Hermitian matrix A. The matrix version of the well-known Kantorovich inequality for a positive definite matrix A is stated as follows (see, e.g., [, ]): for any nonzero vector x ∈ C n . An equivalent form of this result is the inequality valid for any nonzero vector x ∈ C n . This famous inequality plays an important role in statistics (see [, ]; 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 [, ]).
In , 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,  < m < M, such that mI ≤ A ≤ MI in the partial operator order, K(A; x) := Ax, x A - x, x , and C α,β (A) := (A -ᾱI)(βI -A).
A further improvement of the matrix version of (.) is proposed in [], where the classical Kantorovich inequality (.) 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 A ∈ M n with eigenvalues  < λ  ≤ λ  ≤ · · · ≤ λ n : and Then the following inequality holds []: The result above is an improvement of the Kantorovich inequality (.). A generalized form of the Kantorovich inequality presented by Greub and Rheinboldt [] in  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 M i and m i , i = , , respectively. In , Fujii et al.
[] generalized the Greub-Rheinboldt inequality to pairs of invertible operators that may not even commute, By using the viewpoint of interaction antieigenvalue, Gustafson [] sharpened the Greub-Rheinboldt inequality (.) 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 AB = BA with real eigenvalues λ  ≤ λ  ≤ · · · ≤ λ n and μ  ≤ μ  ≤ · · · ≤ μ n , respectively. Moreover, let Ax, Bx := http://www.journalofinequalitiesandapplications.com/content/2013/1/7 (Ax) * Bx = x * A * Bx. Then a matrix version of (.) is for any nonzero vector x ∈ C n . In , Seddighin [] extended the Greub-Rheinboldt inequality (.) to pairs of normal operators and established for what vectors the Greub-Rheinboldt inequality becomes equality.
Let V be an n × r matrix such that V * V = I r , i.e., V is suborthogonal. Another wellknown matrix version of the Kantorovich inequality asserts that Mond and Pečarić proved the following matrix version inequality (see () in []): for A >  and V * V = I. For more related properties and applications, see, e.g., [-].
In the next section, we propose some refinements about the matrix Kantorovich-type inequalities (.), the Greub-Rheinboldt inequality for commutative positive definite Hermitian matrix pairs, and (.) for positive definite matrices, yielding sharpened upper bounds compared with original results, together with an improvement to (.).

Main results
In this section, we first introduce some lemmas.
for any x ∈ C n .
Proof From (.), On the other hand, The proof of D((AB) - , x) is similar.
Proof By Theorem ., we have the following: From the definition of σ  and σ n , we know that α n α  ≥ σ n σ  ≥ . Thus, That is, Remark From Lemma . and (.), we can obtain a sharpened bound for the classical Kantorovich-type inequality, i.e., the Greub-Rheinboldt inequality. http://www.journalofinequalitiesandapplications.com/content/2013/1/7 Besides the discussion on the Greub-Rheinboldt inequality (.), we are also interested in another form of Kantorovich-type inequality aforementioned. We turn our attention to the inequalities (.) and (.) in the remainder of this paper.
Let A be an n × n positive (semi-) definite Hermitian matrix with (nonzero) eigenvalues contained in the interval [m, M], where  < m < M. Let V be n × r matrices.
As is declared in (.), for A > , V * V = I, and m, M mentioned above, the following inequality holds: It is not difficult to see that as V * V = I, then VV * = VV + ≤ I, where + indicates the Moore-Penrose inverse. Multiplying from the right and from the left by V * A and AV respectively, From the well-known Löwner-Heinz inequality, we have (V * A  V ) / ≥ V * AV and the following inequality (see in []): For z ∈ [m, M], m > , the convexity of (z - + z/mM) implies that If A has the representation A = D α * , where is unitary and D α = diag(α  , . . . , α n ), and if  < m ≤ α i ≤ M, i = , . . . , n, then from (.) it follows that After multiplying from the right and from the left by and * , it is not difficult to see that (.) yields the following []: Based on (.), we derive several results on the inequality (.).
Theorem . For any A >  and V * V = I, Since V * V = I, (.) can be turned into By adding (V * A  V ) / ≥  to both sides of the inequality (.), we obtain that i.e., Thus, we finally have Remark It is obvious that D  (A, V ) ≥ . Thus, Theorem . indeed presents an improvement of the Kantorovich-type inequality (.) in [].
For an application to the Hadamard product, we have the following corollary.

Conclusion
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 ., 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. http://www.journalofinequalitiesandapplications.com/content/2013/1/7