Refinements of Kantorovich Inequality for Hermitian Matrices

Abstract

Some new Kantorovich-type inequalities for Hermitian matrix are proposed in this paper. We consider what happens to these inequalities when the positive definite matrix is allowed to be invertible and provides refinements of the classical results.

1. Introduction and Preliminaries

We first state the well-known Kantorovich inequality for a positive definite Hermite matrix (see [1, 2]), let 𝐴∈𝑀_𝑛 be a positive definite Hermitian matrix with real eigenvalues 𝜆₁≤𝜆₂≤⋯≤𝜆_𝑛. Then 1≤𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝜆𝑥≤¹+𝜆_𝑛²4𝜆₁𝜆_𝑛, (1.1) for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1, where 𝐴^∗ denotes the conjugate transpose of matrix 𝐴. A matrix 𝐴∈𝑀_𝑛 is Hermitian if 𝐴=𝐴^∗. An equivalent form of this result is incorporated in 0≤𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝜆𝑥−1≤^𝑛−𝜆₁²4𝜆₁𝜆_𝑛, (1.2) for any 𝑥∈ℂ^𝑛,‖𝑥‖=1.

Attributed to Kantorovich, the inequality has built up a considerable literature. This typically comprises generalizations. Examples are [3–5] for matrix versions. Operator versions are developed in [6, 7]. Multivariate versions have been useful in statistics to assess the robustness of least squares, see [8, 9] and the references therein.

Due to the important applications of the original Kantorovich inequality for matrices [10] in Statistics [8, 11, 12] and Numerical Analysis [13, 14], any new inequality of this type will have a flow of consequences in the areas of applications.

Motivated by the interest in both pure and applied mathematics outlined above we establish in this paper some improvements of Kantorovich inequalities. The classical Kantorovich-type inequalities are modified to apply not only to positive definite but also to invertible Hermitian matrices. As natural tools in deriving the new results, the recent Grüss-type inequalities for vectors in inner product in [6, 15–19] are utilized.

To simplify the proof, we first introduce some lemmas.

2. Lemmas

Let 𝐵 be a Hermitian matrix with real eigenvalues 𝜇₁≤𝜇₂≤⋯≤𝜇_𝑛, if 𝐴−𝐵 is positive semidefinite, we write 𝐴≥𝐵, (2.1) that is, 𝜆_𝑖≥𝜇_𝑖, 𝑖=1,2,…,𝑛. On ℂ^𝑛, we have the standard inner product defined by ⟨𝑥,𝑦⟩=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖, where 𝑥=(𝑥₁,…,𝑥_𝑛)^∗∈ℂ^𝑛 and 𝑦=(𝑦₁,…,𝑦_𝑛)^∗∈ℂ^𝑛.

Lemma 2.1. Let 𝑎, 𝑏, 𝑐, and 𝑑 be real numbers, then one has the following inequality: 𝑎²−𝑏²𝑐²−𝑑²≤(𝑎𝑐−𝑏𝑑)². (2.2)

Lemma 2.2. Let 𝐴 and 𝐵 be Hermitian matrices, if 𝐴𝐵=𝐵𝐴, then ()𝐴𝐵≤𝐴+𝐵²4. (2.3)

Lemma 2.3. Let 𝐴≥0, 𝐵≥0, if 𝐴𝐵=𝐵𝐴, then 𝐴𝐵≥0. (2.4)

3. Some Results

The following lemmas can be obtained from [16–19] by replacing Hilbert space (𝐻,⟨⋅,⋅⟩) with inner product spaces ℂ^𝑛, so we omit the details.

Lemma 3.1. Let 𝑢, 𝑣, and 𝑒 be vectors in ℂ^𝑛, and ‖𝑒‖=1. If 𝛼, 𝛽, 𝛿, and 𝛾 are real or complex numbers such that Re𝛽𝑒−𝑢,𝑢−𝛼𝑒≥0,Re𝛿𝑒−𝑣,𝑣−𝛾𝑒≥0, (3.1) then ⟩−⟨≤1|⟨𝑢,𝑣𝑢,𝑒⟩⟨𝑒,𝑣⟩|4𝛽−𝛼𝛿−𝛾−Re𝛽𝑒−𝑢,𝑢−𝛼𝑒Re𝛿𝑒−𝑣,𝑣−𝛾𝑒^1/2. (3.2)

Lemma 3.2. With the assumptions in Lemma 3.1, one has ⟩−⟨≤1|⟨𝑢,𝑣𝑢,𝑒⟩⟨𝑒,𝑣⟩|4𝛽−𝛼𝛾−𝛿−⟨𝑢,𝑒⟩−𝛼+𝛽2⟨𝑣,𝑒⟩−𝛾+𝛿2. (3.3)

Lemma 3.3. With the assumptions in Lemma 3.1, if Re(𝛽𝛼)≥0, Re(𝛿𝛾)≥0, one has |⟨𝑢,𝑣⟩−⟨𝑢,𝑒⟩⟨𝑒,𝑣⟩|≤𝛽−𝛼𝛿−𝛾4Re𝛽𝛼Re𝛿𝛾^1/2|⟨𝑢,𝑒⟩⟨𝑒,𝑣⟩|. (3.4)

Lemma 3.4. With the assumptions in Lemma 3.3, one has ⟩−⟨≤𝛽|⟨𝑢,𝑣𝑢,𝑒⟩⟨𝑒,𝑣⟩|𝛼+𝛽−2Re𝛼^1/2𝛿+𝛾−2Re𝛿𝛾^1/21/2[|⟨𝑢,𝑒⟩⟨𝑒,𝑣⟩|]^1/2. (3.5)

4. New Kantorovich Inequalities for Hermitian Matrices

For a Hermitian matrix 𝐴, as in [6], we define the following transform: 𝐶(𝐴)=𝜆_𝑛𝐼−𝐴𝐴−𝜆₁𝐼. (4.1) When 𝐴 is invertible, if 𝜆₁𝜆_𝑛>0, then, 𝐶𝐴⁻¹=1𝜆₁𝐼−𝐴⁻¹𝐴⁻¹−1𝜆_𝑛𝐼. (4.2) Otherwise, 𝜆₁𝜆_𝑛<0, then, 𝐶𝐴⁻¹=1𝜆_𝑘+1𝐼−𝐴⁻¹𝐴⁻¹−1𝜆_𝑘𝐼, (4.3) where 𝜆₁≤⋯≤𝜆_𝑘<0<𝜆_𝑘+1≤⋯≤𝜆_𝑛. (4.4)

From Lemma 2.3 we can conclude that 𝐶(𝐴)≥0 and 𝐶(𝐴⁻¹)≥0.

For two Hermitian matrices 𝐴 and 𝐵, and 𝑥∈ℂ^𝑛, ‖𝑥‖=1, we define the following functional: 𝐺(𝐴,𝐵;𝑥)=⟨𝐴𝑥,𝐵𝑥⟩−⟨𝐴𝑥,𝑥⟩⟨𝑥,𝐵𝑥⟩. (4.5) When 𝐴=𝐵, we denote 𝐺(𝐴;𝑥)=‖𝐴𝑥‖²−⟨𝐴𝑥,𝑥⟩², (4.6) for 𝑥∈ℂ^𝑛, ‖𝑥‖=1.

Lemma 4.1. With notations above, and for 𝑥∈ℂ^𝑛, ‖𝑥‖=1, then ⟨𝐶(𝐴)⟩≤𝜆0≤𝑥,𝑥^𝑛−𝜆₁²4, (4.7) if 𝜆_𝑛𝜆₁>0, 0≤𝐶𝐴⁻¹≤𝜆𝑥,𝑥^𝑛−𝜆₁²4𝜆_𝑛𝜆₁², (4.8) if 𝜆_𝑛𝜆₁<0, 0≤𝐶𝐴⁻¹≤𝜆𝑥,𝑥^𝑘+1−𝜆_𝑘²4𝜆_𝑘+1𝜆_𝑘². (4.9)

Proof. From 𝐶(𝐴)≥0, then ⟨𝐶(𝐴)𝑥,𝑥⟩≥0. (4.10) While, from Lemma 2.2, we can get 𝐶(𝐴)=𝜆_𝑛𝐼−𝐴𝐴−𝜆₁𝐼≤𝜆_𝑛−𝜆₁²4𝐼. (4.11) Then ⟨𝐶(𝐴)𝑥,𝑥⟩≤(𝜆_𝑛−𝜆₁)²/4 is straightforward. The proof for 𝐶(𝐴⁻¹) is similar.

Lemma 4.2. With notations above, and for 𝑥∈ℂ^𝑛, ‖𝑥‖=1, then 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤𝐺(𝐴;𝑥)𝐺𝐴⁻¹;𝑥. (4.12)

Proof. Thus, 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²=𝑥^∗𝑥^∗𝐴⁻¹𝑥𝐼−𝐴⁻¹𝑥^∗𝐴𝑥𝐼−𝐴𝑥²≤𝑥^∗𝐴⁻¹𝑥𝐼−𝐴⁻¹𝑥²𝑥^∗𝐴𝑥𝐼−𝐴𝑥², (4.13) while 𝑥^∗𝐴𝑥𝐼−𝐴𝑥²=𝑥^∗𝑥^∗𝐴𝑥²𝐼−2𝑥^∗𝐴𝑥𝐴+𝐴²𝑥=𝑥^∗𝐴²𝑥−𝑥^∗𝐴𝑥²=‖𝐴𝑥‖²−⟨𝐴𝑥,𝑥⟩²=𝐺(𝐴;𝑥). (4.14) Similarly, we can get ‖((𝑥^∗𝐴⁻¹𝑥)𝐼−𝐴⁻¹)𝑥‖²=𝐺(𝐴⁻¹;𝑥), then we complete the proof.

Theorem 4.3. Let 𝐴, 𝐵 be two Hermitian matrices, and 𝐶(𝐴)≥0, 𝐶(𝐵)≥0 are defined as above, then |𝐺(≤1𝐴,𝐵;𝑥)|4𝜆_𝑛−𝜆₁𝜇_𝑛−𝜇₁−[⟨𝐶(𝐴)𝑥,𝑥⟩⟨𝐶(𝐵)𝑥,𝑥⟩]^1/2,|𝐺(≤1𝐴,𝐵;𝑥)|4𝜆_𝑛−𝜆₁𝜇_𝑛−𝜇₁−𝜆𝐴−¹+𝜆_𝑛2𝜇𝑥,𝑥𝐵−¹+𝜇_𝑛2𝑥,𝑥, (4.15) for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1.
If 𝜆₁𝜆_𝑛>0, 𝜇₁𝜇_𝑛>0, then |𝐺(≤𝜆𝐴,𝐵;𝑥)|^𝑛−𝜆₁𝜇_𝑛−𝜇₁4𝜆_𝑛𝜆₁𝜇_𝑛𝜇₁^1/2,|𝐺(≤𝜆|⟨𝐴𝑥,𝑥⟩⟨𝐵𝑥,𝑥⟩|𝐴,𝐵;𝑥)|₁+𝜆_𝑛−2𝜆₁𝜆_𝑛^1/2𝜇₁+𝜇_𝑛−2𝜇₁𝜇_𝑛^1/21/2[|⟨𝐴𝑥,𝑥⟩⟨𝐵𝑥,𝑥⟩|]^1/2, (4.16) for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1.

Proof. The proof follows by Lemmas 3.1, 3.2, 3.3, and 3.4 on choosing 𝑢=𝐴𝑥, 𝑣=𝐵𝑥, and 𝑒=𝑥, 𝛽=𝜆_𝑛, 𝛼=𝜆₁, 𝛿=𝜇_𝑛, and 𝛾=𝜇₁, 𝑥∈ℂ^𝑛, ‖𝑥‖=1, respectively.

Corollary 4.4. Let 𝐴 be a Hermitian matrices, and 𝐶(𝐴)≥0 is defined as above, then |𝐺(≤1𝐴;𝑥)|4𝜆_𝑛−𝜆₁²−⟨𝐶(𝐴)⟩,|𝐺(≤1𝑥,𝑥𝐴;𝑥)|4𝜆_𝑛−𝜆₁²−𝜆𝐴−¹+𝜆_𝑛2𝐼𝑥,𝑥², (4.17)(4.18) for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1.
If 𝜆₁𝜆_𝑛>0, then |𝐺(≤𝜆𝐴;𝑥)|^𝑛−𝜆₁²4𝜆_𝑛𝜆₁|⟨𝐴𝑥,𝑥⟩|²,|𝐺(𝐴;𝑥)|≤𝜆₁+𝜆_𝑛−2𝜆₁𝜆_𝑛|⟨𝐴𝑥,𝑥⟩|, (4.19)(4.20) for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1.

Proof. The proof follows by Theorem 4.3 on choosing 𝐴=𝐵, respectively.

Corollary 4.5. Let 𝐴 be a Hermitian matrices and 𝐶(𝐴⁻¹)≥0 is defined as above, then one has the following.
If 𝜆₁𝜆_𝑛>0, then 𝐺𝐴⁻¹≤𝜆;𝑥^𝑛−𝜆₁²4𝜆_𝑛𝜆₁²−𝐶𝐴⁻¹𝐺𝐴𝑥,𝑥,₋₁≤𝜆;𝑥^𝑛−𝜆₁²4𝜆_𝑛𝜆₁²−𝐴⁻¹−𝜆₁+𝜆_𝑛2𝜆_𝑛𝜆₁𝐼𝑥,𝑥²𝐺𝐴₋₁,≤𝜆;𝑥^𝑛−𝜆₁²4𝜆_𝑛𝜆₁𝐴⁻¹𝑥,𝑥²,𝐺𝐴⁻¹≤𝜆;𝑥¹+𝜆_𝑛𝜆₁𝜆_𝑛1−2𝜆₁𝜆_𝑛𝐴⁻¹𝑥,𝑥, (4.21)(4.22)(4.23)(4.24) for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1.
If 𝜆₁𝜆_𝑛<0, then 𝐺𝐴⁻¹≤𝜆;𝑥^𝑘−𝜆_𝑘+1²4𝜆_𝑘𝜆_𝑘+1²−𝐶𝐴⁻¹𝑥,𝑥, (4.25) where 𝐶(𝐴⁻¹)=((1/𝜆_𝑘+1)𝐼−𝐴⁻¹)(𝐴⁻¹−(1/𝜆_𝑘)𝐼), and 𝐺𝐴⁻¹≤𝜆;𝑥^𝑘−𝜆_𝑘+1²4𝜆_𝑘𝜆_𝑘+1²−𝐴⁻¹−𝜆_𝑘+1+𝜆_𝑘2𝜆_𝑘𝜆_𝑘+1𝐼𝑥,𝑥², (4.26) for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1.

Proof. The proof follows by Corollary 4.4 by replacing 𝐴 with 𝐴⁻¹, respectively.

Theorem 4.6. Let 𝐴 be an 𝑛×𝑛 invertible Hermitian matrix with real eigenvalues 𝜆₁≤𝜆₂≤⋯≤𝜆_𝑛, then one has the following.
If 𝜆₁𝜆_𝑛>0, then 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹≤𝜆𝑥−1^𝑛−𝜆₁²4𝜆_𝑛𝜆₁−⟨𝐶(𝐴)𝑥,𝑥⟩𝐶𝐴⁻¹,𝑥𝑥,𝑥_∗𝐴𝑥𝑥^∗𝐴⁻¹≤𝜆𝑥−1^𝑛−𝜆₁²4𝜆_𝑛𝜆₁−𝜆𝐴−¹+𝜆_𝑛2𝐼𝑥,𝑥𝐴⁻¹−𝜆₁+𝜆_𝑛2𝜆_𝑛𝜆₁𝐼,𝑥𝑥,𝑥_∗𝐴𝑥𝑥^∗𝐴⁻¹≤𝜆𝑥−1^𝑛−𝜆₁²4𝜆_𝑛𝜆₁²⟨𝐴𝑥,𝑥⟩𝐴⁻¹𝑥𝑥,𝑥,_∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1≤𝜆_𝑛−𝜆₁²𝜆_𝑛𝜆₁⟨𝐴𝑥,𝑥⟩𝐴⁻¹𝑥,𝑥, (4.27)(4.28)(4.29)(4.30) for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1.
If 𝜆₁𝜆_𝑛<0, then 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹≤𝜆𝑥−1^𝑛−𝜆₁𝜆_𝑘+1−𝜆_𝑘4𝜆_𝑘𝜆_𝑘+1−⟨𝐶(𝐴)𝑥,𝑥⟩𝐶𝐴⁻¹𝑥,𝑥, (4.31) where 𝐶(𝐴⁻¹)=((1/𝜆_𝑘+1)𝐼−𝐴⁻¹)(𝐴⁻¹−(1/𝜆_𝑘)𝐼), 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹≤𝜆𝑥−1^𝑛−𝜆₁𝜆_𝑘+1−𝜆_𝑘4𝜆_𝑘𝜆_𝑘+1−𝜆𝐴−¹+𝜆_𝑛2𝐼𝑥,𝑥𝐴⁻¹−𝜆_𝑘+𝜆_𝑘+12𝜆₁𝜆_𝑘+1𝐼𝑥,𝑥. (4.32)

Proof. Considering 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤𝐺(𝐴;𝑥)𝐺𝐴⁻¹;𝑥. (4.33) When 𝜆₁𝜆_𝑛>0, from (4.17) and (4.21), we get 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤14𝜆_𝑛−𝜆₁²−⟨𝐶(𝐴)𝜆𝑥,𝑥⟩^𝑛−𝜆₁²4𝜆_𝑛𝜆₁²−𝐶𝐴⁻¹𝑥,𝑥. (4.34) From (𝑎²−𝑏²)(𝑐²−𝑑²)≤(𝑎𝑐−𝑏𝑑)², we have 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤𝜆_𝑛−𝜆₁²4𝜆_𝑛𝜆₁−⟨𝐶(𝐴)𝑥,𝑥⟩𝐶𝐴⁻¹𝑥,𝑥², (4.35) then, the conclusion (4.27) holds.
Similarly, from (4.18) and (4.22), we get 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤14𝜆_𝑛−𝜆₁²−𝜆𝐴−¹+𝜆_𝑛2𝐼𝑥,𝑥²×𝜆_𝑛−𝜆₁²4𝜆_𝑛𝜆₁²−𝐴⁻¹−𝜆₁+𝜆_𝑛2𝜆_𝑛𝜆₁𝐼𝑥,𝑥²≤𝜆_𝑛−𝜆₁²4𝜆_𝑛𝜆₁−𝜆𝐴−¹+𝜆_𝑛2𝐼𝑥,𝑥𝐴⁻¹−𝜆₁+𝜆_𝑛2𝜆_𝑛𝜆₁𝐼𝑥,𝑥², (4.36) then, the conclusion (4.28) holds.
From (4.19) and (4.23), we get 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤𝜆_𝑛−𝜆₁²4𝜆_𝑛𝜆₁|⟨𝐴𝑥,𝑥⟩|²𝜆_𝑛−𝜆₁²4𝜆_𝑛𝜆₁𝐴⁻¹𝑥,𝑥², (4.37) then, the conclusion (4.29) holds.
From (4.20) and (4.24), we get 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤𝜆₁+𝜆_𝑛−2𝜆₁𝜆_𝑛𝜆|⟨𝐴𝑥,𝑥⟩|¹+𝜆_𝑛𝜆₁𝜆_𝑛1−2𝜆₁𝜆_𝑛𝐴⁻¹𝑥,𝑥, (4.38) then, the conclusion (4.30) holds.
When 𝜆₁𝜆_𝑛<0, from (4.17) and (4.25), we get 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤14𝜆_𝑛−𝜆₁²−⟨𝐶(𝐴)𝜆𝑥,𝑥⟩^𝑘−𝜆_𝑘+1²4𝜆_𝑘𝜆_𝑘+1²−𝐶𝐴⁻¹𝑥,𝑥, (4.39) then, the conclusion (4.31) holds.
From (4.18) and (4.26), we get 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1²≤14𝜆_𝑛−𝜆₁²−𝜆𝐴−¹+𝜆_𝑛2𝐼𝑥,𝑥²×𝜆_𝑘−𝜆_𝑘+1²4𝜆_𝑘𝜆_𝑘+1²−𝐴⁻¹−𝜆_𝑘+1+𝜆_𝑘2𝜆_𝑘𝜆_𝑘+1𝐼𝑥,𝑥²≤𝜆_𝑛−𝜆₁𝜆_𝑘+1−𝜆_𝑘4𝜆_𝑘𝜆_𝑘+1−𝜆𝐴−¹+𝜆_𝑛2𝐼𝑥,𝑥𝐴⁻¹−𝜆_𝑘+𝜆_𝑘+12𝜆₁𝜆_𝑘+1𝐼𝑥,𝑥², (4.40) then, the conclusion (4.32) holds.

Corollary 4.7. With the notations above, for any 𝑥∈ℂ^𝑛, ‖𝑥‖=1, one lets 𝐺₁(=1𝐴;𝑥)4𝜆_𝑛−𝜆₁²−⟨𝐶(𝐴)⟩,𝐺𝑥,𝑥₂(=1𝐴;𝑥)4𝜆_𝑛−𝜆₁²−𝜆𝐴−¹+𝜆_𝑛2𝐼𝑥,𝑥². (4.41) If 𝜆₁𝜆_𝑛>0, one lets 𝐺₃(=𝜆𝐴;𝑥)^𝑛−𝜆₁²4𝜆_𝑛𝜆₁|⟨𝐴𝑥,𝑥⟩|²,𝐺₄(𝐴;𝑥)=𝜆₁+𝜆_𝑛−2𝜆₁𝜆_𝑛|⟨𝐴𝑥,𝑥⟩|. (4.42) If 𝜆₁𝜆_𝑛>0𝐺¹𝐴⁻¹=𝜆;𝑥^𝑛−𝜆₁²4𝜆_𝑛𝜆₁²−𝐶𝐴⁻¹,𝐺𝑥,𝑥₂𝐴⁻¹=𝜆;𝑥^𝑛−𝜆₁²4𝜆_𝑛𝜆₁²−𝐴⁻¹−𝜆₁+𝜆_𝑛2𝜆_𝑛𝜆₁𝐼𝑥,𝑥²,𝐺³𝐴⁻¹=𝜆;𝑥^𝑛−𝜆₁²4𝜆_𝑛𝜆₁𝐴⁻¹𝑥,𝑥²,𝐺⁴𝐴⁻¹=𝜆;𝑥¹+𝜆_𝑛𝜆₁𝜆_𝑛1−2𝜆₁𝜆_𝑛𝐴⁻¹𝑥,𝑥. (4.43) If 𝜆₁𝜆_𝑛<0, then 𝐺⁵𝐴⁻¹=𝜆;𝑥^𝑘−𝜆_𝑘+1²4𝜆_𝑘𝜆_𝑘+1²−𝐶𝐴⁻¹𝑥,𝑥, (4.44) where 𝐶(𝐴⁻¹)=((1/𝜆_𝑘+1)𝐼−𝐴⁻¹)(𝐴⁻¹−(1/𝜆_𝑘)𝐼), and 𝐺⁶𝐴⁻¹=𝜆;𝑥^𝑘−𝜆_𝑘+1²4𝜆_𝑘𝜆_𝑘+1²−𝐴⁻¹−𝜆_𝑘+1+𝜆_𝑘2𝜆_𝑘𝜆_𝑘+1𝐼𝑥,𝑥². (4.45) Then, one has the following.
If 𝜆₁𝜆_𝑛>0, 𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1≤𝐺^∗(𝐴;𝑥)𝐺^̇∗𝐴⁻¹;𝑥, (4.46) where 𝐺^∗(𝐴;𝑥)=min𝐺₁(𝐴;𝑥),𝐺₂(𝐴;𝑥),𝐺₃(𝐴;𝑥),𝐺₄(,𝐺𝐴;𝑥)_∗(𝐴;𝑥)=min𝐺¹𝐴⁻¹¹;𝑥,𝐺²𝐴⁻¹;𝑥,𝐺³𝐴⁻¹;𝑥,𝐺⁴𝐴⁻¹;𝑥. (4.47) If 𝜆₁𝜆_𝑛<0𝑥^∗𝐴𝑥𝑥^∗𝐴⁻¹𝑥−1≤𝐺^∘(𝐴;𝑥)𝐺^•𝐴⁻¹;𝑥, (4.48) where 𝐺^∘(𝐴;𝑥)=min𝐺₁(𝐴;𝑥),𝐺₂(,𝐺𝐴;𝑥)_•(𝐴;𝑥)=min𝐺⁵𝐴⁻¹;𝑥,𝐺⁶𝐴⁻¹;𝑥. (4.49)

Proof. The proof follows from that the conclusions in Corollaries 4.4 and 4.5 are independent.

Remark 4.8. It is easy to see that if 𝜆₁>0, 𝜆_𝑛>0, our result coincides with the inequality of operator versions in [6]. So we conclude that our results give an improvement of the Kantorovich inequality [6] that applies to all invertible Hermite matrices.

5. Conclusion

In this paper, we introduce some new Kantorovich-type inequalities for the invertible Hermitian matrices. Inequalities (4.27) and (4.31) are the same as [4], but our proof is simple. In Theorem 4.6, if 𝜆₁>0, 𝜆_𝑛>0, the results are similar to the well-known Kantorovich-type inequalities for operators in [6]. Moreover, for any invertible Hermitian matrix, there exists a similar inequality.

Acknowledgments

The authors are grateful to the associate editor and two anonymous referees whose comments have helped to improve the final version of the present work. This work was supported by the Natural Science Foundation of Chongqing Municipal Education Commission (no. KJ091104).