On the Hermitian positive definite solutions of nonlinear matrix equation X^s + A∗X^−tA = Q

Abstract

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ⩾ 1, 0 < t ⩽ 1 and 0 < s ⩽ 1, t ⩾ 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.

Introduction

We consider the nonlinear matrix equation

$$X^s+A^{\ast }{X}^{-t}A=Q\text{,}$$

where A is a nonsingular complex matrix and Q is a Hermitian positive definite matrix with two cases: s ⩾ 1, 0 < t ⩽ 1 and 0 < s ⩽ 1, t ⩾ 1.

Nonlinear matrix equations with the form of (1.1) have many applications in: dynamic programming; control theory; stochastic filtering; statistics etc., see [1], [4], [18]. The case that s = t = 1 has been extensively studied by several authors [1], [2], [3], [4], [5], [7], [11], [18], [19], and some existence conditions and properties of its positive definite solutions are obtained. Moreover, effective iterative methods for computing the Hermitian positive definite solutions are proposed. Ivanov and his coauthors [12], [13] consider the case that s = 1, t = 2 and Q = I. Liu and Gao [14] did some research on the matrix equation X^s + A∗X^−tA = I with s, t ∈ N, where N denotes the set of natural numbers. Hasanov and Ivanov [8] discuss the Hermitian positive definite solutions of Eq. (1.1) for s = 1, t ∈ N. Yang [17] considers the case s, t ∈ N. In Hasanov [9], [10], the authors investigate the matrix equation when s = 1, t ∈ (0, 1]. Then Peng et al. [16] propose iterative methods for the extremal positive definite solutions of Eq. (1.1) for s = 1 with two cases: 0 < t ⩽ 1 and t ⩾ 1.

In this paper, we discuss the Hermitian positive definite solutions of Eq. (1.1) with more general two cases: s ⩾ 1, 0 < t ⩽ 1 and 0 < s ⩽ 1, t ⩾ 1. The paper is organized as follows. In Section 2, we derive necessary conditions and sufficient conditions for the existence of the Hermitian positive definite solutions for Eq. (1.1) and present some properties of the solutions. Then in Section 3, we propose iterative methods for obtaining the maximal positive definite solution of Eq. (1.1) with the case s ⩾ 1, 0 < t ⩽ 1 and the minimal Hermitian positive definite solution of Eq. (1.1) with the case 0 < s ⩽ 1, t ⩾ 1. We give some numerical examples in Section 4 to show the efficiency of the proposed iterative methods. Conclusions will be put in Section 5.

The following notations are used throughout this paper. Let σ₁(·), σ_n(·) denote the biggest and the smallest singular value of a n × n complex matrix and λ_max(H), λ_min(H) the maximal and the minimal eigenvalue of a Hermitian matrix H, respectively. The notation B ⩾ 0 (B > 0) means that B is a positive semi-definite (definite) matrix. For two Hermitian matrices B and C, the notation B ⩾ C (B > C) indicates that B − C is a positive semi-definite (definite) matrix, and X ∈ [B, C] implies that B ⩽ X ⩽ C. We call a Hermitian positive definite solution X_L of the matrix equation the maximal solution if X_L ⩾ X holds for any Hermitian positive definite solution X of the matrix equation. The minimal solution X_M can be defined similarly. Moreover, A∗ stands for the conjugate transpose of the matrix A and I denotes the identity matrix.