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# Several Matrix Euclidean Norm Inequalities Involving Kantorovich Inequality

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 291984 (2009)

## Abstract

Kantorovich inequality is a very useful tool to study the inefficiency of the ordinary least-squares estimate with one regressor. When regressors are more than one statisticians have to extend it. Matrix, determinant, and trace versions of it have been presented in the literature. In this paper, we provide matrix Euclidean norm Kantorovich inequalities.

## 1. Introduction

Suppose that is an positive definite matrix and is an real vector, then the well-known Kantorovich inequality can be expressed as

where are the eigenvalues of . It is a very useful tool to study the inefficiency of the ordinary least-squares estimate with one regressor in the linear model. Watson [1] introduced the ratio of the variance of the best linear unbiased estimator to the variance of the ordinary least-squares estimator. Such a lower bound of this ratio was provided by Kantorovich inequality (1.1); see, for example, [2, 3]. When regressors are more than one statisticians have to extend it. Marshall and Olkin [4] were the first to generalize Kantorovich inequality to matrices (see, e.g., [5])

where is an real matrix. If , then (1.2) becomes

Bloomfield and Watson [6] and Knott [7] simultaneously established the inequality

where is an real matrix such that and . Yang [8] presented its trace version

where is an real matrix such that .

To the best of our knowledge, there has not been any matrix Euclidean norm version of Kantorovich inequality yet. Our goal is to present its matrix Euclidean norm version.

This paper is arranged as follows. In Section 2, we will give some lemmas which are useful in the following section. In Section 3, some matrix inequalities are established by Kantorovich inequality or Pólya-Szegö inequality, which are referred to as the extensions of Kantorovich inequality as well and conclusions are given in Section 4.

## 2. Some Lemmas

We will start with some lemmas which are very useful in the following.

Definition 2.1.

Let be an complex square matrix. is called a normal matrix if

Lemma 2.2.

Let be an complex square matrix and let be the eigenvalues of , then

where denotes the squared Euclidean norm of . The equality in (2.1) holds if and only if is a normal matrix.

Proof.

See [5].

Lemma 2.3 (Pólya-Szegö inequality).

There is

where .

Moreover Greub and Rheinboldt [9] generalized Pólya-Szegö inequality to matrices.

Lemma 2.4 (Poincare).

Let be an Hermitian matrix, and let be an column orthogonal and full rank matrix, that is , then one has

Let be a unitary matrix, whose column vectors are eigenvectors corresponding to , respectively. Assume and , then if and only if ; while if and only if , where is a unitary matrix.

Proof.

See [5].

## 3. Main Results

Theorem 3.1.

Let and be nonnegative definite Hermitian matrices with , and let be an complex matrix satisfying . Then one has

where and are eigenvalues of matrices and , respectively.

Proof.

We easily get that is a Hermitian matrix since is a Hermitian matrix. Hence is a normal matrix and then we can derive from Lemma 2.2 that

By Lemma 2.4 we get that

Similarly,

Note that

The latter expression of (3.5) may be expressed as

where is an arbitrary permutation of . Clearly, , , and . Therefore, let and , then we can derive from Pólya-Szegö inequality that

Since inequality (3.7) holds for any permutation of thus we find

In the following, the remaining problem is to choose a proper permutation of to minimize

This may be solved by a nontrivial but elementary combinatorial argument, thus we find

Then

Remark 3.2.

When is positive definite Hermitian matrix and , inequality (3.1) plays an important role in the linear model . The covariance matrices of the ordinary least-squares estimator and the best linear unbiased estimator are given in this model

Applying inequality (3.1), we can establish a lower bound of the inefficiency of least-squares estimator

See also [10].

In Theorem 3.1, we need the assumption that . However, we should also point out that the matrix may not meet such an assumption in practice. Therefore, we relax this assumption in the following but the results are weaken.

Theorem 3.3.

Let and be nonnegative definite Hermitian matrices with , and let be an complex matrix, then one has

where and are eigenvalues of matrices and , respectively.

Proof.

If , the result obviously holds. Next set . Let the spectral decomposition of be , where is an orthogonal matrix and Let , then

We can derive from (3.15) that

Similarly,

where We thus have

According to the proof of Theorem 3.1 we can get that

The proof is completed.

Corollary 3.4.

Let be an positive definite Hermitian matrix with eigenvalues , and let be an arbitrary complex matrix, then one has

Proof.

It is very easy to prove therefore we omit the proof.

Theorem 3.5.

Let and be positive definite Hermitian matrices with , and be the eigenvalues of and , respectively, and let be an arbitrary complex matrix. Then,

Proof.

If , the result obviously holds. Next set . Since there exists a unitary matrix such that and , where .

Define . By Corollary 3.4, we can get that

where The right-hand side of (3.22) may be denoted by , then

It is easy to prove that is a momotone increasing function of on interval . Write , then we have From the definitions of and , we thus have

This completes the proof.

Corollary 3.6.

If and are positive semidefinite Hermitian matrices with eigenvalues and , respectively, inequality (3.21) becomes

Theorem 3.7.

Let and be an positive semidefinite Hermitian matrices with , and be the eigenvalues and , respectively, and let , be complex matrices with . Then

Proof.

Note that

Similarly,

Using the abbreviations . Clearly, Let be the eigenvalues of , respectively. Applying Hölder inequality, we can derive that

Thus

This completes the proof.

Corollary 3.8.

When , inequality (3.26) becomes

## 4. Conclusions

The study of the inefficiency of the ordinary least-squares estimator in the linear model requires a lower bound for the efficiency defined as the ratio of the variance or covariance of the best linear unbiased estimator to the variance or covariance of the ordinary least-squares estimator. Such a bound can be given by Kantorovich inequality or its extensions. Matrix, determinant, and trace versions of it have been presented in the literature. In this paper, we present its matrix Euclidean norm version.

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## Acknowledgment

The authors thank very much the associate editors and reviewers for their insightful comments and kind suggestions that lead to improving the presentation.

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Wang, L., Yang, H. Several Matrix Euclidean Norm Inequalities Involving Kantorovich Inequality.
*J Inequal Appl* **2009**, 291984 (2009). https://doi.org/10.1155/2009/291984

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DOI: https://doi.org/10.1155/2009/291984