Skip to main content

Notes on Greub-Rheinboldt inequalities

Abstract

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 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 = ( A ¯ ) 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 A0, if x Ax0, x C n . A is further called positive definite, symbolized A>0, if x Ax>0 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 1 λ 2 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., [1, 2]):

1 x A x x A 1 x ( x x ) 2 ( λ 1 + λ n ) 2 4 λ 1 λ n
(1.1)

for any nonzero vector x C n .

An equivalent form of this result is the inequality

0 x A x x A 1 x ( x x ) 2 1 ( λ 1 λ n ) 2 4 λ 1 λ n
(1.2)

valid for any nonzero vector x C n .

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 [3]) 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 [7],

0K(A;x)1 1 4 ( M m ) 2 m M [ Re C m , M ( A ) x , x Re C 1 m , 1 M ( A 1 ) x , x ] 1 / 2 ,
(1.3)

where A is a self-adjoint bounded linear operator on a complex Hilbert space, 0<m<M, such that mIAMI in the partial operator order, K(A;x):=Ax,x A 1 x,x, and C α , β (A):=(A α ¯ I)(βIA).

A further improvement of the matrix version of (1.3) is proposed in [8], 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 A M n with eigenvalues 0< λ 1 λ 2 λ n :

C(A,x)= x ( λ n IA)(A λ 1 I)x,
(1.4)

and

C ( A 1 , x ) = x ( 1 λ 1 I A 1 ) ( A 1 1 λ n I ) x.
(1.5)

Then the following inequality holds [8]:

0 x Ax x A 1 x1 ( λ 1 λ n ) 2 4 λ 1 λ n C ( A , x ) C ( A 1 , x ) ( λ 1 λ n ) 2 4 λ 1 λ n .
(1.6)

The result above is an improvement of the Kantorovich inequality (1.1).

A generalized form of the Kantorovich inequality presented by Greub and Rheinboldt [1] 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 [9],

Ax,AxBx,Bx ( M 1 M 2 + m 1 m 2 ) 2 4 m 1 m 2 M 1 M 2 A x , B x 2 ,
(1.7)

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=1,2, respectively.

In 1997, Fujii et al. [10] generalized the Greub-Rheinboldt inequality to pairs of invertible operators that may not even commute,

A 2 B 2 x , x A 2 , x 1 / 2 B 2 , x 1 / 2 m 1 m 2 + M 2 M 2 2 m 1 m 2 M 1 M 2 A 2 B 2 x , x A x , B x 2 ,
(1.8)

where A, B are invertible positive operators satisfying 0< m 1 A M 1 and 0< m 2 B M 2 , and AB= A 1 / 2 ( A 1 / 2 B A 1 / 2 ) 1 / 2 A 1 / 2 . By using the viewpoint of interaction antieigenvalue, Gustafson [9] sharpened the Greub-Rheinboldt inequality (1.7) to obtain the following result:

Ax,AxBx,Bx ( m ( A B 1 ) + M ( A B 1 ) ) 2 4 m ( A B 1 ) M ( A B 1 ) A x , B x 2 ,
(1.9)

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 1 λ 2 λ n and μ 1 μ 2 μ n , respectively. Moreover, let Ax,Bx:= ( A x ) Bx= x A Bx. Then a matrix version of (1.9) is

x A 2 x x B 2 x ( x A B x ) 2 ( λ 1 μ 1 + λ n μ n ) 2 4 λ 1 λ n μ 1 μ n
(1.10)

for any nonzero vector x C n .

In 2005, Seddighin [11] 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 n×r matrix such that V V= I r , i.e., V is suborthogonal. Another well-known matrix version of the Kantorovich inequality asserts that

V A 2 V ( m + M ) 2 4 m M ( V A V ) 2
(1.11)

for any A>0, V V=I, and 0<mI<A<MI.

Mond and Pečarić proved the following matrix version inequality (see (7) in [12]):

( V A 2 V ) 1 / 2 V AV ( M m ) 2 4 ( M m ) I
(1.12)

for A>0 and V V=I. For more related properties and applications, see, e.g., [1315].

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 [8], Lemma 2.2)

Let A M n be a positive definite Hermitian matrix. The following inequalities hold:

λ 1 x 2 x Ax λ n x 2 ,0 ( λ n x 2 x A x ) ( x A x λ 1 x 2 ) 1 4 ( λ n λ 1 ) 2 x 4 ,

and

1 λ n x 2 x A 1 x 1 λ 1 x 2 , 0 ( 1 λ 1 x 2 x A 1 x ) ( x A 1 x 1 λ n x 2 ) ( λ n λ 1 ) 2 4 ( λ 1 λ n ) 2 x 4
(2.1)

for any x C n .

Let A, B be two invertible commuting Hermite matrices. Denote by λ 1 λ 2 λ n and μ 1 μ 2 μ n the eigenvalues of A and B, respectively. Then there exists a unitary matrix U M n such that A=UΛ U , B=UM U , where Λ=diag( λ 1 ,, λ n ), M=diag( μ ˆ 1 ,, μ ˆ n ). Note that μ ˆ 1 , μ ˆ 2 ,, μ ˆ n is a permutation of μ 1 , μ 2 ,, μ n . Let σ k = λ k μ ˆ k (k=1,,n), then it is easy to see that all eigenvalues of A B 1 are σ 1 , σ 2 ,, σ n . Without loss of generality, we may assume that σ 1 = min k { λ k μ ˆ k }, σ n = max k { λ k μ ˆ k } and σ 1 σ n . For convenience, we introduce the notation

D(AB,x)= x A ( σ n I A B 1 ) ( A B 1 σ 1 I ) Bx.
(2.2)

If σ 1 σ n >0, then we can define

D ( ( A B ) 1 , x ) = x A ( 1 σ 1 I A 1 B ) ( A 1 B 1 σ n I ) Bx.
(2.3)

Lemma 2.2 Let A and B be two positive definite commuting matrices with eigenvalues 0< λ 1 λ 2 λ n , 0< μ 1 μ 2 μ n , respectively. σ 1 σ 2 σ n , D(AB,x) and D( ( A B ) 1 ,x) are as before. Then for any x C n ,

0 D ( A B , x ) 1 4 ( σ n σ 1 ) 2 | x A B x | , 0 D ( ( A B ) 1 , x ) ( σ n σ 1 ) 2 4 ( σ 1 σ n ) 2 | x A B x |
(2.4)

for any x C n .

Proof From (2.2),

D ( A B , x ) = x A ( σ n I A B 1 ) ( A B 1 σ 1 I ) B x = x U Λ U ( σ n I U Λ U U M 1 U ) ( U Λ U U M 1 U σ 1 I ) U M U x = x U Λ ( σ n I Λ M 1 ) ( Λ M 1 σ 1 I ) M U x .
(2.5)

Let z= ( z 1 , , z n ) T = ( Λ M ) 1 / 2 U x. Thus, z 2 = z z= x U(ΛM) U x= x ABx. Then

D(AB,x)= z ( σ n I Λ M 1 ) ( Λ M 1 σ 1 I ) z= i = 1 n ( σ n σ i )( σ i σ 1 ) z i 2 0.
(2.6)

On the other hand,

i = 1 n ( σ n σ i )( σ i σ 1 ) z i 2 ( σ n σ 1 ) 2 4 z 2 .
(2.7)

Thus,

D(AB,x) ( σ n σ 1 ) 2 4 z 2 = ( σ n σ 1 ) 2 4 | x A B x | .
(2.8)

The proof of D( ( A B ) 1 ,x) is similar. □

Theorem 2.3 With the assumptions of Lemma  2.2,

0 x A 2 x x B 2 x ( x A B x ) 2 1 ( σ n σ 1 ) 2 4 σ 1 σ n 1 | x A B x | D ( A B , x ) D ( ( A B ) 1 , x ) .
(2.9)

Proof Let z= ( Λ M ) 1 / 2 U x, E=Λ M 1 =diag( λ n μ ˆ n ,, λ 1 μ ˆ 1 )=diag( σ n ,, σ 1 ). Then

x A 2 x x B 2 x ( x A B x ) 2 = z E z z E 1 z ( z z ) 2 .
(2.10)

From (1.2) and (1.6),

0 z E z z E 1 z ( z z ) 2 1 ( σ n σ 1 ) 2 4 σ 1 σ n C ( E , z z ) C ( E 1 , z z ) = ( σ n σ 1 ) 2 4 σ 1 σ n 1 z 2 C ( E , z ) C ( E 1 , z ) .
(2.11)

From (2.5) and (2.10), we have

z z= x ABx,C(E,z)=D(AB,x),C ( E 1 , z ) =D ( ( A B ) 1 , x ) .
(2.12)

By substituting (2.12) and (2.10) into (2.11), the inequality becomes

0 x A 2 x x B 2 x ( x A B x ) 2 1 ( σ n σ 1 ) 2 4 σ 1 σ n 1 | x A B x | D ( A B , x ) D ( ( A B ) 1 , x ) .

 □

Corollary 2.4 Let A and B be two positive definite commuting matrices with eigenvalues 0< λ 1 λ n , 0< μ 1 μ n , respectively. Then

x A 2 x x B 2 x ( x A B x ) 2 ( λ 1 μ 1 + λ n μ n ) 2 4 λ 1 μ 1 λ n μ n 1 | x A B x | D ( A B , x ) D ( ( A B ) 1 , x )
(2.13)

holds for any nonzero vector x C n .

Proof

By Theorem 2.3, we have the following:

0 x A 2 x x B 2 x ( x A B x ) 2 ( σ 1 + σ n ) 2 4 σ 1 σ n 1 | x A B x | D ( A B , x ) D ( ( A B ) 1 , x ) .
(2.14)

Let f(x)= ( 1 + x ) 2 4 x . It can be easily deduced that f(x) is monotone increasing on [1,+). Let α 1 = μ 1 λ n , α n = μ n λ 1 . From the definition of σ 1 and σ n , we know that α n α 1 σ n σ 1 1. Thus,

( σ 1 + σ n ) 2 4 σ 1 σ n =f ( σ n σ 1 ) f ( α n α 1 ) = ( λ 1 μ 1 + λ 1 μ 1 ) 2 4 λ 1 μ 1 λ 1 μ 1 .

That is,

0 x A 2 x x B 2 x ( x A B x ) 2 ( λ 1 μ 1 + λ 1 μ 1 ) 2 4 λ 1 μ 1 λ 1 μ 1 1 | x A B x | D ( A B , x ) D ( ( A B ) 1 , x ) .
(2.15)

 □

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 n×n positive (semi-) definite Hermitian matrix with (nonzero) eigenvalues contained in the interval [m,M], where 0<m<M. Let V be n×r matrices.

As is declared in (1.11), for A>0, V V=I, and m, M mentioned above, the following inequality holds:

V A 2 V ( m + M ) 2 4 m M ( V A V ) 2 .

It is not difficult to see that as V V=I, then V V =V V + I, where + indicates the Moore-Penrose inverse. Multiplying from the right and from the left by V A and AV respectively, we have V A 2 V ( V A V ) 2 for A>0. From the well-known Löwner-Heinz inequality, we have ( V A 2 V ) 1 / 2 V AV and the following inequality (see in [16]):

( V A 2 V ) 1 / 2 m + M 2 m M V AV.

For z[m,M], m>0, the convexity of ( z 1 +z/mM) implies that

z 1 m + M m M z m M .
(2.16)

If A has the representation A=Γ D α Γ , where Γ is unitary and D α =diag( α 1 ,, α n ), and if 0<m α i M, i=1,,n, then from (2.16) it follows that

D α 1 m + M m M I D α m M .
(2.17)

After multiplying from the right and from the left by Γ and Γ , it is not difficult to see that (2.17) yields the following [17]:

A 1 m + M m M I A m M .
(2.18)

Based on (2.18), we derive several results on the inequality (1.12).

Theorem 2.5 For any A>0 and V V=I,

( V A 2 V ) 1 / 2 V AV ( M m ) 2 4 ( M + m ) I D 2 (A,V),
(2.19)

where D(A,V)= ( 1 m + M V A 2 V ) 1 / 2 ( M + m ) 1 / 2 2 I.

Proof From (2.18) and A>0, we can get

A m M ( M + m ) I 1 ( M + m ) A 2 .
(2.20)

Since V V=I, (2.20) can be turned into

V AV m M ( M + m ) I 1 ( M + m ) V A 2 V.
(2.21)

By adding ( V A 2 V ) 1 / 2 0 to both sides of the inequality (2.21), we obtain that

( V A 2 V ) 1 / 2 V AV ( V A 2 V ) 1 / 2 m M ( M + m ) I 1 ( M + m ) V A 2 V,
(2.22)

i.e.,

( V A 2 V ) 1 / 2 V A V ( M m ) 2 4 ( M + m ) I 1 ( M + m ) V A 2 V + ( V A 2 V ) 1 / 2 ( M + m ) 4 I = ( M m ) 2 4 ( M + m ) I [ ( 1 M + m V A 2 V ) 1 / 2 ( M + m ) 1 / 2 2 I ] 2 .
(2.23)

Thus, we finally have

( V A 2 V ) 1 / 2 V AV ( m M ) 2 4 ( M + m ) I D 2 (A,V),

where D(A,V)= ( 1 ( m + M ) V A 2 V ) 1 / 2 ( M + m ) 1 / 2 2 I. □

Remark It is obvious that D 2 (A,V)0. Thus, Theorem 2.5 indeed presents an improvement of the Kantorovich-type inequality (1.12) in [12].

For an application to the Hadamard product, we have the following corollary.

Corollary 2.6 Let A 1 and A 2 be n×n positive definite matrices with eigenvalues of A 1 A 2 contained in the interval [m,M]. Then

( A 1 2 A 2 2 ) 1 / 2 A 1 A 2 ( M m ) 2 4 ( m + M ) I D 2 ( A 1 A 2 ,V),

where V is the selection matrix of order n 2 ×n with the property V ( A 1 A 2 )V= A 1 A 2 ( and indicate the tensor and the Hadamard product, respectively).

3 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 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.

References

  1. Greub W, Rheinboldt W: On a generalization of an inequality of L.V. Kantorovich. Proc. Am. Math. Soc. 1959, 10: 407–415. 10.1090/S0002-9939-1959-0105028-3

    Article  MATH  MathSciNet  Google Scholar 

  2. Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985.

    Book  MATH  Google Scholar 

  3. Seddighin M: Antieigenvalue techniques in statistics. Linear Algebra Appl. 2009, 430(10):2566–2580. 10.1016/j.laa.2008.05.007

    Article  MATH  MathSciNet  Google Scholar 

  4. Wang S-G: A matrix version of the Wielandt inequality and its applications to statistics. Linear Algebra Appl. 1999, 296(1–3):171–181. 10.1016/S0024-3795(99)00117-2

    Article  MATH  MathSciNet  Google Scholar 

  5. Householder AS: The Theory of Matrices in Numerical Analysis. Blaisdell, New York; 1964.

    MATH  Google Scholar 

  6. Nocedal J: Theory of algorithms for unconstrained optimization. Acta Numer. 1992, 1: 199–242.

    Article  MathSciNet  Google Scholar 

  7. Dragomir SS: New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. Linear Algebra Appl. 2008, 428(11–12):2750–2760. 10.1016/j.laa.2007.12.025

    Article  MATH  MathSciNet  Google Scholar 

  8. Liu Z, Wang K, Xu C: A note on Kantorovich inequality for Hermite matrices. J. Inequal. Appl. 2011., 2011: Article ID 245767

    Google Scholar 

  9. Gustafson K: Interaction antieigenvalues. J. Math. Anal. Appl. 2004, 299(1):174–185. 10.1016/j.jmaa.2004.06.012

    Article  MATH  MathSciNet  Google Scholar 

  10. Fujii M, Izumino S, Nakamoto R, Seo Y: Operator inequalities related to Cauchy-Schwarz and Holder-McCarthy inequalities. Nihonkai Math. J. 1997, 8(2):117–122.

    MATH  MathSciNet  Google Scholar 

  11. Seddighin M: On the joint antieigenvalue of operators on normal subalgebras. J. Math. Anal. Appl. 2005, 312(1):61–71. 10.1016/j.jmaa.2005.03.021

    Article  MATH  MathSciNet  Google Scholar 

  12. Mond B, Peǎarić JE: A matrix version of the Ky Fan generalization of the Kantorovich inequality. Linear Multilinear Algebra 1994, 36(3):217–221. 10.1080/03081089408818291

    Article  MATH  Google Scholar 

  13. Wang SG, Shao J: Constrained Kantorovich inequalities and relative efficiency of least squares. J. Multivar. Anal. 1992, 42(2):284–298. 10.1016/0047-259X(92)90048-K

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen L, Zeng XM: Rate of convergence of a new type Kantorovich variant of Bleimann-Butzerhahn operators. J. Inequal. Appl. 2009., 2009: Article ID 852897

    Google Scholar 

  15. Yuan JT, Gao ZS: Complete form of Furuta inequality. Proc. Am. Math. Soc. 2008, 136(8):2859–2867. 10.1090/S0002-9939-08-09446-X

    Article  MATH  MathSciNet  Google Scholar 

  16. Liu S, Neudecker H: Several matrix Kantorovich-type inequalities. Aequ. Math. 1990, 40(1):89–93. 10.1007/BF02112284

    Article  MathSciNet  Google Scholar 

  17. Marshall AW, Olkin I: Matrix versions of the Cauchy and Kantorovich inequalities. J. Math. Anal. Appl. 1996, 197(1):23–26. 10.1006/jmaa.1996.0003

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hongyi Li.

Additional information

Competing interests

The authors did not provide this information.

Authors’ contributions

The authors did not provide this information.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-7

Keywords

  • Hilbert Space
  • Matrix Version
  • Hermitian Matrix
  • Nonzero Vector
  • Complex Hilbert Space