Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation 𝑋+∑^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖𝐴_𝑖=𝐼

Abstract

In this paper necessary and sufficient conditions for the matrix equation 𝑋+∑^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖𝐴_𝑖=𝐼 to have a positive definite solution 𝑋 are derived, where −1<𝛿_𝑖<0, 𝐼 is an 𝑛×𝑛 identity matrix, 𝐴_𝑖 are 𝑛×𝑛 nonsingular real matrices, and 𝑚 is an odd positive integer. These conditions are used to propose some properties on the matrices 𝐴_𝑖, 𝑖=1,2,…,𝑚. Moreover, relations between the solution 𝑋 and the matrices 𝐴_𝑖 are derived.

1. Introduction

Consider the nonlinear matrix equation 𝑋+^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖𝐴_𝑖=𝐼,−1<𝛿_𝑖<0, (1) where 𝐼 is an 𝑛×𝑛 identity matrix, 𝐴_𝑖 are 𝑛×𝑛 nonsingular real matrices, and 𝑚 is an odd positive integer (𝐴^∗_𝑖 stands for the conjugate transpose of the matrix 𝐴_𝑖). Several authors [1–13] have studied the existence, the rate of convergence, as well as the necessary and sufficient conditions of the existence of positive definite solutions of similar kinds of nonlinear matrix equations.

This paper is organized as follows. First, in Section 2, we introduce some notations, lemmas, and theorems that will be needed to develop this work. In Section 3, we present necessary and sufficient conditions for the existence of a positive definite solution of (1). In Section 4, some applications of the obtained results as well as relations between the solution 𝑋 and the matrices 𝐴_𝑖 are given.

2. Preliminaries

In this section, we present some notations, lemmas, and theorems that will be needed to develop this paper. We denote by 𝑟(𝐴), 𝜆(𝐴), and ‖𝐴‖ the spectral radius of 𝐴, the eigenvalues of 𝐴, and the Euclidean norm of 𝐴, respectively. The notation 𝐴>𝐵(𝐴≥𝐵) indicates that 𝐴−𝐵 is positive definite (semidefinite).

Lemma 1. Let 𝐸 and 𝐹 be two arbitrary compatible matrices. Then, 𝑟𝐸^𝑇𝐹−𝐹^𝑇𝐸≤𝑟𝐸^𝑇𝐸+𝐹^𝑇𝐹. (2)

The proof of this lemma is given in [14, Lemma 6].

Theorem 2 (Cholesky Decomposition). If 𝐺 is an 𝑛×𝑛 symmetric positive definite matrix, then there exists a triangular 𝑛×𝑛 matrix 𝑀 with positive diagonal entries such that 𝐺=𝑀^𝑇𝑀.

For proof, see [15, page 141].

Theorem 3 (C-S decomposition [15, page 77] and [16, page 37]). If 𝑄 is an orthogonal matrix where 𝑄=^𝑄11𝑄12_𝑄21𝑄22, then there exist orthogonal matrices 𝑈₁, 𝑈₂, 𝑉₁, 𝑉₂ and diagonal matrices Γ≥0, Σ≥0 such that 𝑈𝑄=¹00𝑈₂Γ−Σ𝑉ΣΓ¹00𝑉₂, (3) where Γ²+Σ²=𝐼.

3. Necessary and Sufficient Conditions

In this section, we derive both necessary and sufficient conditions for the existence of a positive definite solution of the nonlinear matrix equation (1).

Theorem 4. The matrix equation (1) has a solution 𝑋 (symmetric and positive definite) if and only if 𝐴_𝑖, 𝑖=1,2,…,𝑚, admit the following factorization: ^𝑚_𝑖=1𝐴_𝑖=^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖, (4) where 𝑊 is a nonsingular square matrix and the columns of 𝑊𝑍₁𝑍₂⋮𝑍_𝑚 (5) are orthonormal. In this case, we take 𝑋=𝑊^𝑇𝑊 as a solution of the matrix equation (1).

Proof. If (1) has a solution 𝑋, then one can write 𝑋 as 𝑋=𝑊^𝑇𝑊 for some nonsingular matrix 𝑊. In particular, 𝑊 can be chosen to be triangular using Theorem 2; thus, we have 𝑋+^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖𝐴_𝑖=𝑊^𝑇𝑊+^𝑚_𝑖=1𝐴^𝑇_𝑖𝑊^𝑇𝑊^𝛿𝑖𝐴_𝑖=𝑊^𝑇𝑊+^𝑚_𝑖=1𝐴^𝑇_𝑖𝑊^𝑇𝑊^𝛿𝑖/2𝑊^𝑇𝑊^𝛿𝑖/2𝐴_𝑖. (6) Then (1) can be rewritten as 𝑊^𝑇𝑊+^𝑚_𝑖=1𝑊^𝑇𝑊^𝛿𝑖/2𝐴_𝑖^𝑇𝑊^𝑇𝑊^𝛿𝑖/2𝐴_𝑖=𝐼, (7) or equivalently 𝑊^𝑇,^𝑚_𝑖=1𝑊^𝑇𝑊^𝛿𝑖/2𝐴_𝑖^𝑇_𝑚𝑊_𝑖∑₌₁𝑊^𝑇𝑊^𝛿𝑖/2𝐴_𝑖=𝐼. (8) Let ^𝑚_𝑖=1𝑊^𝑇𝑊^𝛿𝑖/2𝐴_𝑖=^𝑚_𝑖=1𝑍_𝑖. (9) Then ^𝑚_𝑖=1𝐴_𝑖=^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖, (10) and (8) means that the columns of 𝑊𝑍₁𝑍₂⋮𝑍_𝑚 (11) are orthonormal.
Conversely, suppose that 𝐴_𝑖, 𝑖=1,2,…,𝑚 have the decomposition (4). Set 𝑋=𝑊^𝑇𝑊. Then 𝑋+^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝜹𝑖𝐴_𝑖=𝑊^𝑇𝑊+^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖^𝑇×𝑊^𝑇𝑊^𝛿𝑖𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖=𝑊^𝑇𝑊+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑊^𝑇𝑊^{−𝛿𝑖/2}×𝑊^𝑇𝑊^𝛿𝑖/2𝑊^𝑇𝑊^𝛿𝑖/2×𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖=𝑊^𝑇𝑊+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖=𝐼. (12) That is, 𝑋 is a solution to the matrix equation (1).

Theorem 5. The matrix equation (1) has a solution if and only if there exist orthogonal matrices 𝑃 and 𝑈₂ and diagonal matrices Γ>0 and Σ≥0 with ((𝑚+1)/2)(Γ²+Σ²)=𝐼 such that ^𝑚_𝑖=1𝐴_𝑖=^𝑚_(𝑖odd)𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1(𝑖even)}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1Γ𝑃. (13) In this case, 𝑋=𝑃^𝑇Γ²𝑃 is a solution of (1).

Proof. Suppose that the matrix equation (1) has a solution. Then from Theorem 4,   𝐴_𝑖, 𝑖=1,2,…,𝑚, have the form ^𝑚_𝑖=1𝐴_𝑖=^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖. (14)
Since 𝑊𝑍₁𝑍₂⋮𝑍_𝑚 (15)
is column-orthonormal, it can be extended to an orthogonal matrix 𝑍𝑊𝑈₁𝑉₁𝑍₂𝑉₂𝑍⋮⋮_𝑚𝑉_𝑚, (16)
and applying Theorem 3, there exist orthogonal matrices 𝑈₁,𝑈₂,…,𝑈_𝑚+1,𝑃,𝑄 and diagonal matrices Γ>0 and Σ≥0 such that 𝑍𝑊𝑈₁𝑉₁𝑍₂𝑉₂𝑍⋮⋮_𝑚𝑉_𝑚=𝑈₁00𝑈₂𝑈⋮⋮_𝑚00𝑈_𝑚+1,Γ−ΣΣΓ𝑃00𝑄 (17) where ((𝑚+1)/2)(Γ²+Σ²)=𝐼.
This decomposition is a generalization for Theorem 3 in [15, page 77] and [16, page 37]. It is clear that if we set 𝑚=1, we get the special case of that theorem.
Thus 𝑊=𝑈₁Γ𝑃, and ^𝑚_𝑖=1𝑍_𝑖=^𝑚_𝑖=1,3𝑈_𝑖+1Σ𝑃+^𝑚−1_𝑖=2,4𝑈_𝑖+1Γ𝑃. (18)
Then (4) can be rewritten as follows: ^𝑚_𝑖=1𝐴_𝑖=^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖=^𝑚_𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1Γ𝑃. (19)
Conversely, suppose that 𝐴_𝑖, 𝑖=1,2,…,𝑚 have the decomposition (13). Let 𝑋=𝑃^𝑇Γ²𝑃. Then 𝑋+^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖𝐴_𝑖=𝑈₁Γ𝑃^𝑇𝑈₁+Γ𝑃_𝑚𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1Σ𝑃^𝑇𝑃^𝑇Γ²𝑃^𝛿𝑖×𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1Γ𝑃^𝑇𝑃^𝑇Γ²𝑃^𝛿𝑖×𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1Γ𝑃=𝑃^𝑇Γ²𝑃+𝑃^𝑇Γ²𝑃^−𝛿1/2𝑈₂Σ𝑃^𝑇×𝑃^𝑇Γ²𝑃^𝛿1𝑃^𝑇Γ²𝑃^−𝛿1/2𝑈₂+𝑃Σ𝑃_𝑇Γ²𝑃^−𝛿3/2𝑈₄Σ𝑃^𝑇𝑃^𝑇Γ²𝑃^𝛿3×𝑃^𝑇Γ²𝑃^−𝛿3/2𝑈₄𝑃Σ𝑃+⋯+^𝑇Γ²𝑃^{−𝛿𝑚/2}𝑈_𝑚+1Σ𝑃^𝑇×𝑃^𝑇Γ²𝑃^𝛿𝑚𝑃^𝑇Γ²𝑃^{−𝛿𝑚/2}𝑈_𝑚+1+𝑃Σ𝑃_𝑇Γ²𝑃^−𝛿2/2𝑈₃Γ𝑃^𝑇𝑃^𝑇Γ²𝑃^𝛿2×𝑃^𝑇Γ²𝑃^−𝛿2/2𝑈₃𝑃Γ𝑃+⋯+^𝑇Γ²𝑃^{−𝛿𝑚−1/2}𝑈_𝑚Γ𝑃^𝑇×𝑃^𝑇Γ²𝑃^𝛿𝑚−1𝑃^𝑇Γ²𝑃^{−𝛿𝑚−1/2}𝑈_𝑚Γ𝑃=𝑃^𝑇Γ²𝑃+𝑃^𝑇Σ𝑈^𝑇₂𝑃^𝑇Γ²𝑃^−𝛿1/2×𝑃^𝑇Γ²𝑃^𝛿1𝑃^𝑇Γ²𝑃^−𝛿1/2𝑈₂+𝑃Σ𝑃_𝑇Σ𝑈^𝑇₄𝑃^𝑇Γ²𝑃^−𝛿3/2×𝑃^𝑇Γ²𝑃^𝛿3𝑃^𝑇Γ²𝑃^−𝛿3/2𝑈₄𝑃Σ𝑃+…+^𝑇Σ𝑈^𝑇_𝑚+1𝑃^𝑇Γ²𝑃^{−𝛿𝑚/2}×𝑃^𝑇Γ²𝑃^𝛿𝑚𝑃^𝑇Γ²𝑃^{−𝛿𝑚/2}𝑈_𝑚+1+𝑃Σ𝑃_𝑇Γ𝑈^𝑇₃𝑃^𝑇Γ²𝑃^−𝛿2/2×𝑃^𝑇Γ²𝑃^𝛿2𝑃^𝑇Γ²𝑃^−𝛿2/2𝑈₃+𝑃Γ𝑃_𝑇Γ𝑈^𝑇_𝑚𝑃^𝑇Γ²𝑃^{−𝛿𝑚−1/2}×𝑃^𝑇Γ²𝑃^𝛿𝑚−1𝑃^𝑇Γ²𝑃^{−𝛿𝑚−1/2}𝑈_𝑚Γ𝑃=𝑃^𝑇Γ²𝑃+𝑃^𝑇Σ𝑈^𝑇₂𝑈₂Σ𝑃+𝑃^𝑇Σ𝑈^𝑇₄𝑈₄Σ𝑃+⋯+𝑃^𝑇Σ𝑈^𝑇_𝑚+1𝑈_𝑚+1Σ𝑃+𝑃^𝑇Γ𝑈^𝑇₃𝑈₃Γ𝑃+⋯+𝑃^𝑇Γ𝑈^𝑇_𝑚𝑈_𝑚Γ𝑃=𝑃^𝑇Γ²𝑃+𝑃^𝑇Σ²𝑃+𝑃^𝑇Σ²𝑃+⋯+𝑃^𝑇Σ²𝑃+𝑃^𝑇Γ²𝑃+⋯+𝑃^𝑇Γ²𝑃=𝑃^𝑇𝑚+12Γ²+Σ²𝑃=𝑃^𝑇𝑃=I. (20) This shows that 𝑋 is a solution to the matrix equation (1).

4. Main Results

In this section, we explain some properties of (1) and we obtain relations between the solution 𝑋 and the matrices 𝐴_𝑖.

Theorem 6. If the matrix equation (1) has a positive definite solution 𝑋, then ‖∑^𝑚_𝑖=1𝐴_𝑖‖<𝑚.

Proof. Suppose that (1) has a solution. Then by Theorem 5, ∑^𝑚_𝑖=1𝐴_𝑖 has the decomposition (13).
Then ^𝑚_𝑖=1𝐴_𝑖=^𝑚_𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1≤Γ𝑃_𝑚𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1≤Γ𝑃_𝑚𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1Σ𝑃‖+‖‖‖_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1‖Γ‖‖𝑃‖. (21) Since each 𝑈_𝑖+1, 𝑖=1,2,…,𝑚, is orthogonal, also 𝑃 is orthogonal. So ^𝑚_𝑖=1𝐴_𝑖≤^𝑚_𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}Σ‖+_{𝑚−1𝑖=2,4}‖𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}Γ‖≤_𝑚𝑖=1,3‖𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}‖Σ‖+^𝑚−1_𝑖=2,4𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}‖Γ‖=^𝑚_𝑖=1,3Γ^{2−𝛿𝑖/2}‖Σ‖+^𝑚−1_𝑖=2,4Γ^{2−𝛿𝑖/2}‖Γ‖≤^𝑚_𝑖=1,3‖Γ‖^−𝛿𝑖‖Σ‖+^𝑚−1_𝑖=2,4‖Γ‖^−𝛿𝑖‖Γ‖. (22) Since we choose −1<𝛿_𝑖<0, then we have ‖Γ‖^−𝛿𝑖≤‖Γ‖<1, and therefore ^𝑚_𝑖=1,3‖Γ‖^−𝛿𝑖≤𝑚+12,^𝑚−1_𝑖=2,4‖Γ‖^−𝛿𝑖≤𝑚−12. (23) Thus, ^𝑚_𝑖=1𝐴_𝑖≤𝑚+122+𝑚+1𝑚−122𝑚+1=𝑚2𝑚+1<𝑚. (24) The last inequality follows from the fact ((𝑚+1)/2)(Γ²+Σ²)=𝐼, which yields that ‖Γ‖≤2/(𝑚+1) and ‖Σ‖<2/(𝑚+1).
In the following theorem, using the necessary and sufficient conditions which we have derived, we state and prove some inequalities in case of the matrix equation (1) having positive definite solution.

Theorem 7. Suppose that the matrix equation (1) has a positive definite solution 𝑋; then the following hold:(i)∑^𝑚_𝑖=1𝑋^−𝛿𝑖>∑^𝑚_𝑖=1𝐴_𝑖𝐴^𝑇_𝑖,(ii)𝐼−∑^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖+1𝐴_𝑖−∑^𝑚_𝑖=1𝑋^{(𝛿𝑖+1)/2}𝐴_𝑖𝐴^𝑇_𝑖𝑋^{(𝛿𝑖+1)/2}>0,(iii)𝑟(∑^𝑚_𝑖=1𝐴_𝑖)≤(1/2)2/(𝑚+1)((3𝑚−1)/(𝑚+1)),(iv)𝑟(∑^𝑚_𝑖=1𝑋^{(1+𝛿𝑖)/2}𝐴_𝑖−∑^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^{(1+𝛿𝑖)/2})≤1,(v)𝑟(∑^𝑚_𝑖=1𝑋^{(1+𝛿𝑖)/2}𝐴_𝑖+∑^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^{(1+𝛿𝑖)/2})≤1.

Proof. (i) By Theorem 4, we have 𝑋=𝑊^𝑇𝑊 and ∑^𝑚_𝑖=1𝐴_𝑖=∑^𝑚_𝑖=1(𝑊^𝑇𝑊)^{−𝛿𝑖/2}𝑍_𝑖. Then ^𝑚_𝑖=1𝑋^−𝛿𝑖−^𝑚_𝑖=1𝐴_𝑖𝐴^𝑇_𝑖=^𝑚_𝑖=1𝑊^𝑇𝑊^−𝛿𝑖−^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖𝑍^𝑇_𝑖𝑊^𝑇𝑊^{−𝛿𝑖/2}=^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑊^𝑇𝑊^{−𝛿𝑖/2}−^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖𝑍^𝑇_𝑖𝑊^𝑇𝑊^{−𝛿𝑖/2}=^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝐼−^𝑚_𝑖=1𝑍_𝑖𝑍^𝑇_𝑖𝑊^𝑇𝑊^{−𝛿𝑖/2}>0, (25)
since 𝜆(𝑍_𝑖𝑍^𝑇_𝑖)=𝜆(𝑍^𝑇_𝑖𝑍_𝑖), 𝑖=1,2,…,𝑚, and 𝐼−∑^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖=𝑊^𝑇𝑊>0, and, hence, 𝐼−∑^𝑚_𝑖=1𝑍_𝑖𝑍^𝑇_𝑖>0. Thus, part (i) is proved.
(ii) Replacing 𝐼 by 𝑊^𝑇𝑊+∑^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖, we get 𝐼−^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖+1𝐴_𝑖−^𝑚_𝑖=1𝑋^{(𝛿𝑖+1)/2}𝐴_𝑖𝐴^𝑇_𝑖𝑋^{(𝛿𝑖+1)/2}=𝑊^𝑇𝑊+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖−^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖+1𝐴_𝑖−^𝑚_𝑖=1𝑋^{(𝛿𝑖+1)/2}𝐴_𝑖𝐴^𝑇_𝑖𝑋^{(𝛿𝑖+1)/2}=𝑊^𝑇𝑊+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖−^𝑚_𝑖=1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖^𝑇×𝑊^𝑇𝑊^𝛿𝑖+1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖−^𝑚_𝑖=1𝑊^𝑇𝑊^{(𝛿𝑖+1)/2}𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖×𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖^𝑇𝑊^𝑇𝑊^{(𝛿𝑖+1)/2}=𝑊^𝑇𝑊+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖−^𝑚_𝑖=1𝑍^𝑇_𝑖𝑊^𝑇𝑊^{−𝛿𝑖/2}×𝑊^𝑇𝑊^𝛿𝑖+1𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖−^𝑚_𝑖=1𝑊^𝑇𝑊^{(𝛿𝑖+1)/2}𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖×𝑍^𝑇_𝑖𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑊^𝑇𝑊^{(𝛿𝑖+1)/2}=𝑊^𝑇𝑊^1/2𝑊^𝑇𝑊^1/2+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖−^𝑚_𝑖=1𝑍^𝑇_𝑖𝑊^𝑇𝑊𝑍_𝑖−𝑊^𝑇𝑊^𝑚1/2_𝑖=1𝑍_𝑖𝑍^𝑇_𝑖𝑊^𝑇𝑊^1/2=𝑊^𝑇𝑊^1/2𝐼−^𝑚_𝑖=1𝑍_𝑖𝑍^𝑇_𝑖×𝑊^𝑇𝑊^1/2+^𝑚_𝑖=1𝑍^𝑇_𝑖𝐼−𝑊^𝑇𝑊𝑍_𝑖>0. (26)
This completes the proof of (ii).
(iii) Using Theorem 6, we get 𝜆^𝑚_𝑖=1𝐴_𝑖=𝜆^𝑚_𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1,𝑟Γ𝑃_𝑚𝑖=1𝐴_𝑖=max_𝑖𝜆^𝑚_𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1≤Γ𝑃_𝑚𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1≤Γ𝑃_𝑚𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1+Σ𝑃_{𝑚−1𝑖=2,4}𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}𝑈_𝑖+1=Γ𝑃_𝑚𝑖=1,3𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}Σ+^𝑚−1_𝑖=2,4𝑃^𝑇Γ²𝑃^{−𝛿𝑖/2}Γ. (27)
Since we choose −1<𝛿_𝑖<0, then we have 0<(−𝛿_𝑖)/2<1/2<1, and therefore (𝑃^𝑇Γ²𝑃)^{−𝛿𝑖/2}≤(𝑃^𝑇Γ²𝑃), 𝑖=1,2,…,𝑚; then 𝑟^𝑚_𝑖=1𝐴_𝑖≤𝑚+12𝑃^𝑇Γ²𝑃Σ+𝑚−12𝑃^𝑇Γ²𝑃Γ=𝑚+12𝑃^𝑇ΣΓ²𝑃+𝑚−12𝑃^𝑇Γ³𝑃=𝑚+12ΣΓ²+𝑚−12Γ³≤𝑚+12‖ΣΓΓ‖+𝑚−12‖Γ‖³≤𝑚+12‖ΣΓ‖‖Γ‖+𝑚−12‖Γ‖³. (28)
Let Σ=diag(𝜎_𝑗), Γ=diag(𝛾_𝑗). Then ((𝑚+1)/2)(𝜎²_𝑗+𝛾²_𝑗)=1. Thus 𝑟^𝑚_𝑖=1𝐴_𝑖≤𝑚+122‖‖+𝑚+1ΣΓ𝑚−122𝑚+1³=𝑚+122𝑚+1max_𝑗𝜎_𝑗𝛾_𝑗+𝑚−122𝑚+1³≤𝑚+122𝑚+1max_𝑗𝜎²_𝑗+𝛾²_𝑗2+𝑚−122𝑚+1³=122𝑚+13𝑚−1.𝑚+1 (29)
(iv) Appling Lemma 1, we get 𝑟^𝑚_𝑖=1𝑋^{(1+𝛿𝑖)/2}𝐴_𝑖−^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^{(1+𝛿𝑖)/2}=𝑟^𝑚_𝑖=1𝑊^𝑇𝑊^{(1+𝛿𝑖)/2}𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑍_𝑖−^𝑚_𝑖=1𝑍^𝑇_𝑖𝑊^𝑇𝑊^{−𝛿𝑖/2}𝑊^𝑇𝑊^{(1+𝛿𝑖)/2}=𝑟^𝑚_𝑖=1𝑊^𝑇𝑊^1/2𝑍_𝑖−^𝑚_𝑖=1𝑍^𝑇_𝑖𝑊^𝑇𝑊^1/2≤𝑟𝑊^𝑇𝑊+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖=𝑟(𝐼)=1. (30)
(v) Consider 𝐼±^𝑚_𝑖=1𝑋^{(1+𝛿𝑖)/2}𝐴_𝑖+^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^{(1+𝛿𝑖)/2}=𝑊^𝑇𝑊+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑍_𝑖±^𝑚_𝑖=1𝑊^𝑇𝑊^1/2𝑍_𝑖+^𝑚_𝑖=1𝑍^𝑇_𝑖𝑊^𝑇𝑊^1/2=𝑊^𝑇𝑊^1/2±^𝑚_𝑖=1𝑍_𝑖^𝑇𝑊^𝑇𝑊^1/2±^𝑚_𝑖=1𝑍_𝑖≥0. (31) Therefore, 𝑟(∑^𝑚_𝑖=1𝑋^{(1+𝛿𝑖)/2}𝐴_𝑖+∑^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^{(1+𝛿𝑖)/2})≤1.

5. Conclusion

In this paper, both necessary and sufficient conditions for the nonlinear matrix equation 𝑋+∑^𝑚_𝑖=1𝐴^𝑇_𝑖𝑋^𝛿𝑖𝐴_𝑖=𝐼 to have a positive definite solution 𝑋 are given, where −1<𝛿_𝑖<0, 𝐴_𝑖 are 𝑛×𝑛 nonsingular real matrices, and 𝑚 is an odd positive integer. Some properties of this matrix equation are explained. Also, relations between the positive definite solution 𝑋 and the matrices 𝐴_𝑖, 𝑖=1,2,…,𝑚, are presented.