Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations

Abstract

We study the Hermitian positive definite solutions of the nonlinear matrix equation 𝑋+𝐴^∗𝑋⁻²𝐴=𝐼, where 𝐴 is an 𝑛×𝑛 nonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations of 𝑋+𝐴^∗𝑋⁻²𝐴=𝐼 are presented while the matrix equation has a Hermitian positive definite solution.

1. Introduction

In this paper, we conside the Hermitian positive definite solutions of the matrix equation: 𝑋+𝐴^∗𝑋⁻²𝐴=𝐼, (1.1) where 𝐴 is an 𝑛×𝑛 nonsingular matrix, 𝐼 denotes the identity matrix, and 𝐴^∗ the conjugate transpose of 𝐴.

In many physical applications [1] (such as super-resolution image restoration, a new image processing branch, which restores one better image of higher resolution from a degraded image sequence of lower resolution, algorithms of super-resolution image restoration can be used wherever image resolution is not contented, and image sequence can be obtained), we must solve a system of linear equations [2]: 𝑀𝑥=𝑓, (1.2) where the positive definite matrix 𝑀 arises from a finite different approximation to an elliptic partial differential equation. The solving of the System 𝑀𝑥=𝑓 can be transformed to the solving of the equations 𝑋±𝐴^∗𝑋⁻²𝐴=𝐼, and the nonlinear matrix equation (1.1), appears in such many applications including control theory, ladder networks, dynamic programming, stochastic filtering, statistic, and super-resolution image restoration. For such decomposition to exist the matrix 𝑋 must be a positive definite solution to the matrix equations 𝑋+𝐴^∗𝑋⁻²𝐴=𝐼.

We know that the study of the Hermitian positive definite solutions of the matrix equation has three basic problems, the theoretic issue on solvability; the numerical solution; the analysis of the perturbation.

Because of the wide mathematical and physical background, in recent years, the existence of Hermitian positive solutions for nonlinear matrix equation received wide attention (1.1) [3], for various values of 𝑞. In [3], Ivanov et al. construct iterative methods for obtaining positive definite solutions of the matrix equation (1.1) and give some sufficient conditions for the existence of a positive definite solution. Moreover, Ferrante and Levy [4] studied the equation 𝑋=𝑄+𝑁𝑋⁻¹𝑁^∗, and presented an algorithm which converges to the positive definite solution for a wide range of conditions. Cheng [5] presented some sufficient conditions and new necessary conditions for the existence of Hermitian positive definite solutions. In [6], Ivanov discussed sufficient conditions for existence the minimal 𝑋_𝑠 and special 𝑋_𝑙 positive definite solutions are derived and iterative procedures for computing these solutions. The problem in the Hermitian positive definite solutions of the matrix equation has been extensively researched in [7–12].

Throughout this paper, let ‖⋅‖ denote the spectral norm for square matrices unless otherwise noted, that is, ‖𝐴‖=max_𝑖𝜆_𝑖, where the 𝜆_𝑖 are the eigenvalues of 𝐴𝐴^∗, 𝐼 the identity matrix, and 𝐴^∗ the conjugate transpose of 𝐴. The notation 𝑋>0 means that 𝑋 is a positive definite Hermitian matrix, 𝐴>𝐵 is used to indicate that 𝐴−𝐵>0, that is, 𝐴−𝐵 be Hermitian positive definite.

In this paper, motivated by the results mentioned earlier, we give some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of (1.1), and these necessary and sufficient conditions are different to [3, 5]. Based on them, we also present some properties of the coefficient matrix 𝐴 are presented and two equivalent equations of (1.1) when the matrix equation has a Hermitian positive definite solution.

2. Preliminaries and Lemmas

The following are several basic properties and are useful in this paper.

Lemma 2.1. The spectral norm is monotonic norm, that is, if 0<𝐴≤𝐵, then ‖𝐴‖≤‖𝐵‖.

Lemma 2.2. If 𝑀 is positive definite Hermitian matrix, then there is a unique positive definite Hermitian matrix 𝑊 such that 𝑀=𝑊².

Lemma 2.3 (see [3, Corollary  2.1]). If 𝑋 is a positive definite solution of (1.1), then 𝐼<𝑋<𝐼+𝐴^∗𝐴. (2.1)

Lemma 2.4 (see [13, Theorem  4]). If (1.1) has a solution, then ‖𝐴‖<1. (2.2) Moreover, if 𝐴 has order >2, then ‖𝐴‖ can take any value in the interval [0,1) for (1.1) to have a solution.

3. The Main Results

In this section, we present our main results.

Theorem 3.1. The matrix equation (1.1) has a Hermitian positive definite solution 𝑋 if and only if 𝐴 unitary equivalent to Λ²𝑍, that is, 𝐴 can be factorized as 𝐴=𝑈^∗Λ²𝑍𝑈, (3.1) where Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑠), 𝜆_𝑖 are the eigenvalues of 𝑋, 𝑖=1,2,…,𝑛, 𝑋=𝑈^∗Λ²𝑈, and the columns of Λ𝑍 (3.2) are orthonormal. In this case 𝑋=𝑈^∗Λ²𝑈 is a Hermitian positive definite solution of (1.1) and all Hermitian positive definite solutions of (1.1) can be formed in this way. By the result mentioned earlier, the solving of (1.1) can be transformed to the solving of the equation Λ₁+𝐶^∗Λ₁⁻²𝐶=𝐼, (3.3) where 𝐶=𝑈𝐴𝑈^∗ is a nonsingular Hermitian matrix, Λ₁=diag(𝜆₁,𝜆₂,…,𝜆_𝑠), 𝜆_𝑖 are the eigenvalues of 𝑋, 𝑖=1,2,…,𝑛.

Proof. If the matrix equation (1.1) has a Hermitian positive definite solution 𝑋, then there exists a unitary matrix 𝑈 satisfied 𝑋=𝑈^∗Λ₁𝑈 where Λ₁=diag(𝜆₁,𝜆₂,…,𝜆_𝑛),𝜆_𝑖(𝑖=1,2,…,𝑛) are the eigenvalues of 𝑋. Let Λ₁=Λ², then Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑠), we rewrite the matrix equation (1.1) as 𝑈^∗Λ²𝑈+𝐴^∗(𝑈^∗Λ²𝑈)⁻²𝐴=𝐼, (3.4) so 𝑈^∗Λ²𝑈+𝐴^∗𝑈^∗Λ⁻⁴𝑈𝐴=𝐼. (3.5) Because 𝑈 is a unitary matrix, hence Λ²+(𝑈^∗)⁻¹𝐴^∗𝑈^∗Λ⁻⁴𝑈𝐴𝑈⁻¹=(𝑈^∗)⁻¹𝐼𝑈⁻¹=𝐼. (3.6) Thus ΛΛ⁻²𝑈𝐴𝑈^∗∗ΛΛ⁻²𝑈𝐴𝑈^∗=𝐼. (3.7) Let 𝑍=Λ⁻²𝑈𝐴𝑈^∗, then 𝐴=𝑈^∗Λ²𝑍𝑈 and by (3.3), it is easy to see that the columns of Λ𝑍 (3.8) are orthonormal.
Conversely, suppose that 𝐴 has the decomposition (3.1) and 𝑋=𝑈^∗Λ²𝑈, where 𝑈 is a unitary matrix. Since the columns of Λ𝑍 (3.9) are orthonormal, so ΛΛ⁻²𝑈𝐴𝑈^∗∗ΛΛ⁻²𝑈𝐴𝑈^∗=𝐼, (3.10) that is, Λ²+(𝑈^∗)⁻¹𝐴^∗𝑈^∗Λ⁻⁴𝑈𝐴𝑈⁻¹=(𝑈^∗)⁻¹𝐼𝑈⁻¹=𝐼. (3.11)
Then, we have 𝑋+𝐴^∗𝑋⁻²𝐴=𝑈^∗Λ²𝑈+𝐴^∗(𝑈^∗Λ𝑈)⁻⁴𝐴=𝑈^∗Λ𝑈+(𝑈^∗Λ²𝑍𝑈)^∗𝑈^∗Λ⁻⁴𝑈𝑈^∗Λ²𝑍𝑈=𝐼. (3.12)
Hence 𝑋 is a Hermitian positive definite solution of (1.1).
The proof of Theorem 3.1 is complete.

Theorem 3.2. If the matrix equation (1.1) has a Hermitian positive definite solution 𝑋, then ‖𝐴‖^1/2<𝜆<1,𝜆<1−‖𝐴‖, (3.13) where 𝜆=max{𝜆_𝑖,𝑖=1,2,…,𝑛},𝜆=min{𝜆_𝑖,𝑖=1,2,…,𝑛},𝜆_𝑖,𝑖=1,2,…,𝑛 are the eigenvalues of 𝑋.

Proof. By Theorem 3.1, if Λ₁ is a Hermitian positive definite solution of (3.3), then we have Λ₁>0,𝐶^∗Λ₁⁻²𝐶>0, (3.14) thus 𝐶^∗Λ₁⁻²𝐶=𝐼−Λ₁>0, (3.15) hence 𝐼−Λ₁>0, that is, 0<𝜆_𝑖<1, (3.16) where 𝜆_𝑖,𝑖=1,2,…,𝑛.
Without loss of generality, suppose 0<𝜆₁<𝜆₂<⋯<𝜆_𝑛<1, then 𝜆=𝜆_𝑛<1. By the proof of Theorem 3.1, we have (Λ₁⁻¹𝑈𝐴)^∗(Λ₁⁻¹𝑈𝐴)=𝑈^∗(𝐼−Λ₁)𝑈. (3.17) Let ∑=diag(𝜎₁,𝜎₂,…,𝜎_𝑛)≥0 such that Λ₁+∑²=𝐼, where 𝜎₁≥𝜎₂≥⋯≥𝜎_𝑛. Since 𝜆<1, then ∑=diag(𝜎₁,𝜎₂,…,𝜎_𝑛)>0, 1>𝜎₁≥𝜎₂≥⋯≥𝜎_𝑛>0, and 𝜎₁=1−𝜆₁, that is, 𝜎₁=1−𝜆. (3.18)
Now let 𝐵=Λ₁⁻¹𝑈𝐴𝑈^∗=(𝑏₁,𝑏₂,…,𝑏_𝑛), (3.19) where 𝑏_𝑖 is 𝑛-column vector, then 𝐵^∗𝐵=∑, so 𝐵 is column orthogonal and 𝑏^∗_𝑖𝑏_𝑖=𝜎²_𝑖,𝑖=1,2,…,𝑛. Let 𝑣_𝑖=𝑏_𝑖/𝜎_𝑖,𝑖=1,2,…,𝑛, and with CS factorization theorem, extend them to form an orthogonal basis of 𝐶^𝑛. Thus 𝑉=(𝑣₁,𝑣₂,,𝑣_𝑛) is a unitary matrix such that 𝐵=𝑉∑, and hence 𝐴=𝑈^∗Λ₁𝑉𝑈. (3.20) So ‖𝐴‖=‖𝑈^∗Λ₁𝑉𝑈‖=‖Λ₁𝑉‖≤‖Λ₁=‖‖‖𝜆²(1−𝜆), (3.21) thus we have ‖𝐴‖≤𝜆², (3.22) that is, ‖𝐴‖^1/2≤𝜆,‖𝐴‖≤1−𝜆, (3.23) that is, 𝜆≤1−‖𝐴‖. (3.24) The proof of Theorem 3.2 is complete.

Theorem 3.3. If the matrix equation (1.1) has a Hermitian positive definite solution 𝑋, let 𝛼=max{𝜆²_𝑖,𝑖=1,2,…,𝑛},𝛽=max{1−𝜆_𝑖,𝑖=1,2,…,𝑛},𝜆=min{𝜆_𝑖,𝑖=1,2,…,𝑛}, where 𝜆_𝑖,𝑖=1,2,…,𝑛 are the eigenvalues of 𝑋, then(1)𝐴^∗𝐴<𝛽Λ²<𝛽𝛼𝐼;(2)𝜆𝐼<Λ<𝐼−𝛼𝐴^∗𝐴.

Proof. By Theorem 3.1, 𝐶=𝑈𝐴𝑈^∗, where 𝑈 is a unitary matrix, hence 𝐴^∗𝐴=𝑈𝐶^∗𝐶𝑈^∗.
Let 𝛼=max{𝜆²_𝑖,𝑖=1,2,…,𝑛},𝛽=max{1−𝜆_𝑖,𝑖=1,2,…,𝑛},𝜆=min{𝜆_𝑖,𝑖=1,2,…,𝑛}, where 𝜆_𝑖,𝑖=1,2,…,𝑛 be the eigenvalues of 𝑋.
(1) By Theorem 3.1, Λ₁ is a Hermitian positive definite solution of (3.3), thus Λ<𝛼𝐼,𝐶^∗Λ₁⁻²𝐶=𝐼−Λ₁<𝐼, (3.25) hence 𝐶^∗Λ₁⁻²𝐶=(𝐼−Λ₁)<𝛽𝐼,Λ²₁<𝛼𝐼<𝐼. (3.26)
So 𝐴^∗𝐴=𝑈^∗𝐶^∗𝐶𝑈<𝛽𝑈^∗Λ²₁𝑈<𝛼𝛽𝐼. (3.27)
(2) Since Λ₁ is a Hermitian positive definite solution of (3.3), 𝐶 is a nonsingular matrix, that is, 𝐶^∗Λ₁⁻²𝐶=𝐼−Λ₁, by (1) Λ₁⁻²>𝛼𝐼, then 𝐶^∗Λ₁⁻²𝐶>𝛼𝐶^∗𝐶, (3.28) so 𝛼𝐴^∗𝐴=𝑈𝛼𝐶^∗𝐶𝑈^∗<𝐼−Λ₁, (3.29) thus we have 𝜆𝐼<Λ₁<𝐼−𝛼𝐴^∗𝐴. (3.30)
The proof of Theorem 3.3 is complete.

Theorem 3.4. Suppose 𝐴 and 𝐵 are two positive definite Hermitian matrices, and 𝐴>𝐵, then there exist a nonsingular matrices 𝑇 which satisfied(1)𝑇^∗𝐴𝑇=𝐼,𝑇^∗𝐵𝑇=Λ, (3.31) where Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛), 𝜆_𝑖 are the eigenvalues of 𝐴⁻¹𝐵,𝑖=1,2,…,𝑛;(2)0<𝜆_𝑖<1, and the columns of Λ^1/2(𝐴−𝐵)^1/2𝑇 (3.32) are orthonormal, 𝑖=1,2,…,𝑛.

Proof. (1) Since the matrix 𝐴 is positive definite Hermitian, then there exists a unique positive definite Hermitian matrix 𝑊 such that 𝐴=𝑊². We note 𝐹=(𝑊⁻¹)^∗𝐵𝑊⁻¹=𝑊𝐵𝑊⁻¹. (3.33) It is to prove that 𝐹 is Hermitian matrix also, so there exists a unitary matrix 𝑈 such that 𝑈^∗𝐹𝑈=Λ, where Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛), 𝜆_𝑖 are the eigenvalues of 𝐹. Hence we have the following equation 𝑈^∗(𝑊⁻¹)^∗𝐵𝑊⁻¹𝑈=Λ. (3.34) Let 𝑇=𝑊⁻¹𝑈, then 𝑇 is nonsingular. Since det(𝜆𝐼−𝐹)=det(𝑊(𝜆𝐼−𝐹)𝑊)=det(𝐴)det(𝜆𝑊^∗𝑊−𝑊^∗𝐹𝑊)=det(𝐴)det(𝜆𝐴−𝐵)=det(𝐴)det(𝜆𝐴−𝐴𝐴⁻¹𝐵)det(𝐴)=det(𝜆𝐼−𝐴⁻¹𝐵), (3.35) so the eigenvalue of 𝐴⁻¹𝐵 is equal to the eigenvalue of 𝐹.
(2) If (3.31) is hold, then we have 𝑇^∗(𝐴−𝐵)𝑇=𝐼−Λ. (3.36) Since 𝐴>𝐵, 𝑇 is nonsingular, so 𝑇^∗(𝐴−𝐵)𝑇 is positive definite Hermitian.
Thus 𝐼−Λ>0, that is, 0<𝜆_𝑖<1,𝑖=1,2,…,𝑛. It is easy to see that the columns of Λ^1/2(𝐴−𝐵)^1/2𝑇 (3.37) are orthonormal.
The proof of Theorem 3.4 is complete.

Theorem 3.5. The matrix equation (1.1) has a Hermitian positive definite solution 𝑋 if and only if 𝐴 has the following factorization: 𝐴=𝑇Λ^−1/2𝑍, (3.38) where Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛),𝜆_𝑖,𝑖=1,2,…,𝑛 are the eigenvalues of 𝑋⁻³,𝑇^∗𝑋𝑇=𝐼,𝑇^∗𝑋⁻²𝑇=Λ and the columns of 𝑇⁻¹𝑍 (3.39) are orthonormal. In this case 𝑋=(𝑇⁻¹)^∗𝑇⁻¹ is a Hermitian positive definite solution and all Hermitian positive definite solutions can be formed in this way.

Proof. If the matrix equation (1.1) has a Hermitian positive definite solution 𝑋, then by Theorem 3.3, there exists a nonsingular matrix 𝑇 which satisfied 𝑇^∗𝑋𝑇=𝐼,𝑇^∗𝑋⁻²𝑇=Λ, (3.40) hence 𝑋=(𝑇⁻¹)^∗𝑇⁻¹,𝑋⁻²=(𝑇⁻¹)^∗Λ𝑇⁻¹. (3.41)
So (1.1) is equivalent to (𝑇⁻¹)^∗𝑇⁻¹+𝐴^∗(𝑇⁻¹)^∗Λ𝑇⁻¹𝐴=𝐼, (3.42) or equivalently 𝑇⁻¹Λ^1/2𝑇⁻¹𝐴^∗𝑇⁻¹Λ^1/2𝑇⁻¹𝐴=𝐼. (3.43) Let Λ^1/2𝑇⁻¹𝐴=𝑍, then 𝐴=𝑇Λ^−1/2𝑍 and (3.43) means that the columns of 𝑇⁻¹Λ^1/2𝑇⁻¹𝐴 (3.44) are orthonormal.
Conversely, suppose that 𝐴 has the decomposition 𝐴=𝑇Λ^−1/2𝑍 and 𝑋=(𝑇⁻¹)^∗𝑇⁻¹, where 𝑈 is a unitary matrix. Since the columns of 𝑇⁻¹𝑍 (3.45) are orthonormal, so 𝑇⁻¹Λ^1/2𝑇⁻¹𝐴^∗𝑇⁻¹Λ^1/2𝑇⁻¹𝐴=𝐼. (3.46) Hence (𝑇⁻¹)^∗𝑇⁻¹+𝐴^∗(𝑇⁻¹)^∗Λ𝑇⁻¹𝐴=𝐼. (3.47) Then we have 𝑋+𝐴^∗𝑋⁻²𝐴=(𝑇⁻¹)^∗𝑇⁻¹+(𝑇⁻¹)^∗Λ𝑇⁻¹=𝐼. (3.48) The proof of Theorem 3.5 is complete.

Theorem 3.6. The matrix equation (1.1) has a Hermitian positive definite solution 𝑋 if and only if 𝐴 has the following factorization: 𝐴=𝑇𝑍, (3.49) where Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛),𝜆_𝑖,𝑖=1,2,…,𝑛 are the eigenvalues of 𝑋³,𝑇^∗𝑋𝑇=Λ,𝑇^∗𝑋⁻²𝑇=𝐼 and the columns of Λ^1/2𝑇⁻¹𝑍 (3.50) are orthonormal. In this case 𝑋=(𝑇⁻¹)^∗Λ𝑇⁻¹ is a Hermitian positive definite solution and all Hermitian positive definite solutions can be formed in this way.

Proof. If the matrix equation (1.1) has a Hermitian positive definite solution 𝑋, then by Theorem 3.3, there exists a nonsingular matrix 𝑇 which satisfied 𝑇^∗𝑋𝑇=Λ,𝑇^∗𝑋⁻²𝑇=𝐼, (3.51) hence 𝑋=(𝑇⁻¹)^∗Λ𝑇⁻¹,𝑋⁻²=(𝑇⁻¹)^∗𝑇⁻¹. (3.52)
So (1.1) is equivalent to (𝑇⁻¹)^∗Λ𝑇⁻¹+𝐴^∗(𝑇⁻¹)^∗𝑇⁻¹𝐴=𝐼, (3.53) or equivalently Λ^1/2𝑇⁻¹𝑇⁻¹𝐴^∗Λ^1/2𝑇⁻¹𝑇⁻¹𝐴=𝐼. (3.54) Let 𝑇⁻¹𝐴=𝑍, then 𝐴=𝑇𝑍, and (3.54) means that the columns of Λ^1/2𝑇⁻¹𝑇⁻¹𝐴 (3.55) are orthonormal.
Conversely, suppose that 𝐴 has the decomposition 𝐴=𝑇𝑍 and 𝑋=(𝑇⁻¹)^∗Λ𝑇⁻¹, where 𝑈 is a unitary matrix. Since the columns of Λ^1/2𝑇⁻¹𝑍 (3.56) are orthonormal, then Λ^1/2𝑇⁻¹𝑇⁻¹𝐴^∗Λ^1/2𝑇⁻¹𝑇⁻¹𝐴=𝐼. (3.57) Thus (𝑇⁻¹)^∗Λ𝑇⁻¹+𝐴^∗(𝑇⁻¹)^∗𝑇⁻¹𝐴=𝐼. (3.58) Then we have 𝑋+𝐴^∗𝑋⁻²𝐴=(𝑇⁻¹)^∗Λ𝑇⁻¹+(𝑇⁻¹)^∗𝑇⁻¹=𝐼. (3.59) The proof of Theorem 3.6 is complete.

Theorem 3.7. The matrix equation (1.1) has a Hermitian positive definite solution 𝑋 if and only if Λ𝑇 is unitary, where Λ=Λ₁+𝐼,Λ₁=diag(𝜆₁,𝜆₂,…,𝜆_𝑛),𝜆_𝑖,𝑖=1,2,…,𝑛 are the eigenvalues of 𝑋⁻¹𝐴^∗𝑋⁻²𝐴.

Proof. If the matrix equation (1.1) has a Hermitian positive definite solution 𝑋, then by Theorem 3.3, there exists a nonsingular matrix 𝑇 which satisfied 𝑇^∗𝑋𝑇=𝐼,𝑇^∗𝐴^∗𝑋⁻²𝐴𝑇=Λ₁, (3.60) hence 𝑋=(𝑇⁻¹)^∗𝑇⁻¹,𝐴^∗𝑋⁻²𝐴=(𝑇⁻¹)^∗Λ₁𝑇⁻¹. (3.61)
So (1.1) is equivalent to (𝑇⁻¹)^∗𝑇⁻¹+(𝑇⁻¹)^∗Λ₁𝑇⁻¹=𝐼, (3.62) that is, (𝑇⁻¹)^∗(𝐼+Λ₁)𝑇⁻¹=𝐼. (3.63) Let Λ=Λ₁+𝐼, then (3.63) is equivalent to (𝑇⁻¹)^∗Λ²𝑇⁻¹=𝐼. (3.64) So (Λ𝑇⁻¹)^∗(Λ𝑇⁻¹)=𝐼. (3.65) Thus Λ𝑇 is unitary.
The proof of Theorem 3.7 is complete.

Similarly, we have the following theorem.

Theorem 3.8. The matrix equation (1.1) has a Hermitian positive definite solution 𝑋 if and only if Λ𝑇 is unitary, where Λ=Λ₁+𝐼,Λ₁=diag(𝜆₁,𝜆₂,…,𝜆_𝑛),𝜆_𝑖,𝑖=1,2,…,𝑛 are the eigenvalues of 𝐴⁻¹𝑋²(𝐴⁻¹)^∗𝑋.

Remark 3.9. By Theorems 3.7 and 3.8, the solving of (1.1) is transformed to the solving of 𝑇^∗𝑇=Λ, (3.66) where Λ=diag(𝜆₁−1,𝜆₂−1,…,𝜆_𝑛)−1,𝜆_𝑖,𝑖=1,2,…,𝑛 are the eigenvalues of 𝐴⁻¹𝑋²(𝐴⁻¹)^∗𝑋 or 𝑋⁻¹𝐴^∗𝑋⁻²𝐴 is a nonsingular Hermitian matrix.
It is easy to see that every eigenvalue 𝜆_𝑖 of 𝐴⁻¹𝑋²(𝐴⁻¹)^∗𝑋 or 𝑋⁻¹𝐴^∗𝑋⁻²𝐴 satisfied 𝜆_𝑖>1. (3.67)

Acknowledgment

This research was supported by KJM of Shandong University of Technology Grant (2005KJM30).