# Refinements of Kantorovich Inequality for Hermitian Matrices

Authors: Feixiang Chen
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 1. Introduction and Preliminaries

We first state the well-known Kantorovich inequality for a positive definite Hermite matrix (see [1, 2]), let A∈Mn be a positive definite Hermitian matrix with real eigenvalues λ1≤λ2≤⋯≤λn. Then
(1.1)1≤x*Axx*A-1x≤(λ1+λn)24λ1λn,
for any x∈ℂn, ∥x∥=1, where A* denotes the conjugate transpose of matrix A. A matrix A∈Mn is Hermitian if A=A*. An equivalent form of this result is incorporated in
(1.2)0≤x*Axx*A-1x-1≤(λn-λ1)24λ1λn,
for any x∈ℂn,∥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

LetB be a Hermitian matrix with real eigenvalues μ1≤μ2≤⋯≤μn, if A-B is positive semidefinite, we write
(2.1)A≥B,
that is, λi≥μi, i=1,2,…,n. On ℂn, we have the standard inner product defined by 〈x,y〉=∑i=1n‍xiy-i, where x=(x1,…,xn)*∈ℂn and y=(y1,…,yn)*∈ℂn.Lemma 2.1.
Leta, b, c, and d be real numbers, then one has the following inequality:
(2.2)(a2-b2)(c2-d2)≤(ac-bd)2.Lemma 2.2.
LetA and B be Hermitian matrices, if AB=BA, then
(2.3)AB≤(A+B)24.Lemma 2.3.
LetA≥0, B≥0, if AB=BA, then
(2.4)AB≥0.


## 3. Some Results

The following lemmas can be obtained from [16–19] by replacing Hilbert space (H,〈·,·〉) with inner product spaces ℂn, so we omit the details.Lemma 3.1.
Letu, v, and e be vectors in ℂn, and ∥e∥=1. If α, β, δ, and γ are real or complex numbers such that
(3.1)Re〈βe-u,u-αe〉≥0,Re〈δe-v,v-γe〉≥0,
then
(3.2)|〈u,v〉-〈u,e〉〈e,v〉|≤14(β-α)(δ-γ)-[Re〈βe-u,u-αe〉Re〈δe-v,v-γe〉]1/2.Lemma 3.2.
With the assumptions in Lemma3.1, one has
(3.3)|〈u,v〉-〈u,e〉〈e,v〉|≤14(β-α)(γ-δ)-|〈u,e〉-α+β2||〈v,e〉-γ+δ2|.Lemma 3.3.
With the assumptions in Lemma3.1, if Re(βα-)≥0, Re(δγ-)≥0, one has
(3.4)|〈u,v〉-〈u,e〉〈e,v〉|≤|β-α||δ-γ|4[Re(βα-)Re(δγ-)]1/2|〈u,e〉〈e,v〉|.Lemma 3.4.
With the assumptions in Lemma3.3, one has
(3.5)|〈u,v〉-〈u,e〉〈e,v〉|≤{(|α+β|-2[Re(βα-)]1/2)(|δ+γ|-2[Re(δγ-)]1/2)}1/2[|〈u,e〉〈e,v〉|]1/2.


## 4. New Kantorovich Inequalities for Hermitian Matrices

For a Hermitian matrixA, as in [6], we define the following transform:
(4.1)C(A)=(λnI-A)(A-λ1I).
When A is invertible, if λ1λn>0, then,
(4.2)C(A-1)=(1λ1I-A-1)(A-1-1λnI).
Otherwise, λ1λn<0, then,
(4.3)C(A-1)=(1λk+1I-A-1)(A-1-1λkI),
where
(4.4)λ1≤⋯≤λk<0<λk+1≤⋯≤λn.From Lemma2.3 we can conclude that C(A)≥0 and C(A-1)≥0.For two Hermitian matricesA and B, and x∈ℂn, ∥x∥=1, we define the following functional:
(4.5)G(A,B;x)=〈Ax,Bx〉-〈Ax,x〉〈x,Bx〉.
When A=B, we denote
(4.6)G(A;x)=∥Ax∥2-〈Ax,x〉2,
for x∈ℂn, ∥x∥=1.Lemma 4.1.
With notations above, and forx∈ℂn, ∥x∥=1, then
(4.7)0≤〈C(A)x,x〉≤(λn-λ1)24,
if λnλ1>0,
(4.8)0≤〈C(A-1)x,x〉≤(λn-λ1)24(λnλ1)2,
if λnλ1<0,
(4.9)0≤〈C(A-1)x,x〉≤(λk+1-λk)24(λk+1λk)2.Proof.
FromC(A)≥0, then
(4.10)〈C(A)x,x〉≥0.
While, from Lemma 2.2, we can get
(4.11)C(A)=(λnI-A)(A-λ1I)≤(λn-λ1)24I.
Then 〈C(A)x,x〉≤(λn-λ1)2/4 is straightforward. The proof for C(A-1) is similar.Lemma 4.2.
With notations above, and forx∈ℂn, ∥x∥=1, then
(4.12)|x*Axx*A-1x-1|2≤G(A;x)G(A-1;x).Proof.
Thus,(4.13)|x*Axx*A-1x-1|2=|x*((x*A-1x)I-A-1)((x*Ax)I-A)x|2≤∥((x*A-1x)I-A-1)x∥2∥((x*Ax)I-A)x∥2,
while
(4.14)∥((x*Ax)I-A)x∥2=x*((x*Ax)2I-2(x*Ax)A+A2)x=x*A2x-(x*Ax)2=∥Ax∥2-〈Ax,x〉2=G(A;x).
Similarly, we can get ∥((x*A-1x)I-A-1)x∥2=G(A-1;x), then we complete the proof.Theorem 4.3.
LetA, B be two Hermitian matrices, and C(A)≥0, C(B)≥0 are defined as above, then
(4.15)|G(A,B;x)|≤14(λn-λ1)(μn-μ1)-[〈C(A)x,x〉〈C(B)x,x〉]1/2,|G(A,B;x)|≤14(λn-λ1)(μn-μ1)-|〈(A-λ1+λn2)x,x〉||〈(B-μ1+μn2)x,x〉|,
for any x∈ℂn, ∥x∥=1.
Ifλ1λn>0, μ1μn>0, then
(4.16)|G(A,B;x)|≤(λn-λ1)(μn-μ1)4[(λnλ1)(μnμ1)]1/2|〈Ax,x〉〈Bx,x〉|,|G(A,B;x)|≤{(|λ1+λn|-2(λ1λn)1/2)(|μ1+μn|-2(μ1μn)1/2)}1/2[|〈Ax,x〉〈Bx,x〉|]1/2,
for any x∈ℂn, ∥x∥=1.Proof.
The proof follows by Lemmas3.1, 3.2, 3.3, and 3.4 on choosing u=Ax, v=Bx, and e=x, β=λn, α=λ1, δ=μn, and γ=μ1, x∈ℂn, ∥x∥=1, respectively.Corollary 4.4.
LetA be a Hermitian matrices, and C(A)≥0 is defined as above, then
(4.17)|G(A;x)|≤14(λn-λ1)2-〈C(A)x,x〉,(4.18)|G(A;x)|≤14(λn-λ1)2-|〈(A-λ1+λn2I)x,x〉|2,
for any x∈ℂn, ∥x∥=1.
Ifλ1λn>0, then
(4.19)|G(A;x)|≤(λn-λ1)24(λnλ1)|〈Ax,x〉|2,(4.20)|G(A;x)|≤(|λ1+λn|-2λ1λn)|〈Ax,x〉|,
for any x∈ℂn, ∥x∥=1.Proof.
The proof follows by Theorem4.3 on choosing A=B, respectively.Corollary 4.5.
LetA be a Hermitian matrices and C(A-1)≥0 is defined as above, then one has the following.
Ifλ1λn>0, then
(4.21)|G(A-1;x)|≤(λn-λ1)24(λnλ1)2-〈C(A-1)x,x〉,(4.22)|G(A-1;x)|≤(λn-λ1)24(λnλ1)2-|〈(A-1-λ1+λn2λnλ1I)x,x〉|2,(4.23)|G(A-1;x)|≤(λn-λ1)24λnλ1|〈A-1x,x〉|2,(4.24)|G(A-1;x)|≤(|λ1+λn|λ1λn-21λ1λn)|〈A-1x,x〉|,
for any x∈ℂn, ∥x∥=1.
Ifλ1λn<0, then
(4.25)|G(A-1;x)|≤(λk-λk+1)24(λkλk+1)2-〈C(A-1)x,x〉,
where C(A-1)=((1/λk+1)I-A-1)(A-1-(1/λk)I), and
(4.26)|G(A-1;x)|≤(λk-λk+1)24(λkλk+1)2-|〈(A-1-λk+1+λk2λkλk+1I)x,x〉|2,
for any x∈ℂn, ∥x∥=1.Proof.
The proof follows by Corollary4.4 by replacing A with A-1, respectively.Theorem 4.6.
LetA be an n×n invertible Hermitian matrix with real eigenvalues λ1≤λ2≤⋯≤λn, then one has the following.
Ifλ1λn>0, then
(4.27)|x*Axx*A-1x-1|≤(λn-λ1)24(λnλ1)-〈C(A)x,x〉〈C(A-1)x,x〉,(4.28)|x*Axx*A-1x-1|≤(λn-λ1)24(λnλ1)-|〈(A-λ1+λn2I)x,x〉||〈(A-1-λ1+λn2λnλ1I)x,x〉|,(4.29)|x*Axx*A-1x-1|≤(λn-λ1)24(λnλ1)2〈Ax,x〉〈A-1x,x〉,(4.30)|x*Axx*A-1x-1|≤(|λn|-|λ1|)2λnλ1〈Ax,x〉〈A-1x,x〉,
for any x∈ℂn, ∥x∥=1.
Ifλ1λn<0, then
(4.31)|x*Axx*A-1x-1|≤(λn-λ1)(λk+1-λk)4|λkλk+1|-〈C(A)x,x〉〈C(A-1)x,x〉,
where C(A-1)=((1/λk+1)I-A-1)(A-1-(1/λk)I),
(4.32)|x*Axx*A-1x-1|≤(λn-λ1)(λk+1-λk)4|λkλk+1|-|〈(A-λ1+λn2I)x,x〉||〈(A-1-λk+λk+12λ1λk+1I)x,x〉|.Proof.
Considering(4.33)|x*Axx*A-1x-1|2≤G(A;x)G(A-1;x).
When λ1λn>0, from (4.17) and (4.21), we get
(4.34)|x*Axx*A-1x-1|2≤{14(λn-λ1)2-〈C(A)x,x〉}{(λn-λ1)24(λnλ1)2-〈C(A-1)x,x〉}.
From (a2-b2)(c2-d2)≤(ac-bd)2, we have
(4.35)|x*Axx*A-1x-1|2≤{(λn-λ1)24(λnλ1)-[〈C(A)x,x〉〈C(A-1)x,x〉]}2,
then, the conclusion (4.27) holds.
Similarly, from (4.18) and (4.22), we get
(4.36)|x*Axx*A-1x-1|2≤{14(λn-λ1)2-|〈(A-λ1+λn2I)x,x〉|2}×{(λn-λ1)24(λnλ1)2-|〈(A-1-λ1+λn2λnλ1I)x,x〉|2}≤{(λn-λ1)24(λnλ1)-|〈(A-λ1+λn2I)x,x〉||〈(A-1-λ1+λn2λnλ1I)x,x〉|}2,
then, the conclusion (4.28) holds.
From (4.19) and (4.23), we get
(4.37)|x*Axx*A-1x-1|2≤(λn-λ1)24(λnλ1)|〈Ax,x〉|2(λn-λ1)24(λnλ1)|〈A-1x,x〉|2,
then, the conclusion (4.29) holds.
From (4.20) and (4.24), we get
(4.38)|x*Axx*A-1x-1|2≤(|λ1+λn|-2λ1λn)|〈Ax,x〉|(|λ1+λn|λ1λn-21λ1λn)|〈A-1x,x〉|,
then, the conclusion (4.30) holds.
Whenλ1λn<0, from (4.17) and (4.25), we get
(4.39)|x*Axx*A-1x-1|2≤{14(λn-λ1)2-〈C(A)x,x〉}{(λk-λk+1)24(λkλk+1)2-〈C(A-1)x,x〉},
then, the conclusion (4.31) holds.
From (4.18) and (4.26), we get
(4.40)|x*Axx*A-1x-1|2≤{14(λn-λ1)2-|〈(A-λ1+λn2I)x,x〉|2}×{(λk-λk+1)24(λkλk+1)2-|〈(A-1-λk+1+λk2λkλk+1I)x,x〉|2}≤{(λn-λ1)(λk+1-λk)4|λkλk+1|-|〈(A-λ1+λn2I)x,x〉||〈(A-1-λk+λk+12λ1λk+1I)x,x〉|}2,
then, the conclusion (4.32) holds.Corollary 4.7.
With the notations above, for anyx∈ℂn, ∥x∥=1, one lets
(4.41)|G1(A;x)|=14(λn-λ1)2-〈C(A)x,x〉,|G2(A;x)|=14(λn-λ1)2-|〈(A-λ1+λn2I)x,x〉|2.
If λ1λn>0, one lets
(4.42)|G3(A;x)|=(λn-λ1)24(λnλ1)|〈Ax,x〉|2,|G4(A;x)|=(|λ1+λn|-2λ1λn)|〈Ax,x〉|.
If λ1λn>0(4.43)|G1(A-1;x)|=(λn-λ1)24(λnλ1)2-〈C(A-1)x,x〉,|G2(A-1;x)|=(λn-λ1)24(λnλ1)2-|〈(A-1-λ1+λn2λnλ1I)x,x〉|2,|G3(A-1;x)|=(λn-λ1)24λnλ1|〈A-1x,x〉|2,|G4(A-1;x)|=(|λ1+λn|λ1λn-21λ1λn)|〈A-1x,x〉|.
If λ1λn<0, then
(4.44)|G5(A-1;x)|=(λk-λk+1)24(λkλk+1)2-〈C(A-1)x,x〉,
where C(A-1)=((1/λk+1)I-A-1)(A-1-(1/λk)I), and
(4.45)|G6(A-1;x)|=(λk-λk+1)24(λkλk+1)2-|〈(A-1-λk+1+λk2λkλk+1I)x,x〉|2.
Then, one has the following.
Ifλ1λn>0,
(4.46)|x*Axx*A-1x-1|≤G*(A;x)G*˙(A-1;x),
where
(4.47)G*(A;x)=min{G1(A;x),G2(A;x),G3(A;x),G4(A;x)},G*(A;x)=min{G1(A-11;x),G2(A-1;x),G3(A-1;x),G4(A-1;x)}.
If λ1λn<0(4.48)|x*Axx*A-1x-1|≤G∘(A;x)G•(A-1;x),
where
(4.49)G∘(A;x)=min{G1(A;x),G2(A;x)},G•(A;x)=min{G5(A-1;x),G6(A-1;x)}.Proof.
The proof follows from that the conclusions in Corollaries4.4 and 4.5 are independent.Remark 4.8.
It is easy to see that ifλ1>0, λn>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, λn>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.



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