The extremal solution of the matrix equation $X^s+A^{\ast }{X}^{-q}A=I$

Abstract

In this paper, the distribution of the positive definite solution of the nonlinear equation $X^s+A^{\ast }{X}^{-q}A=I$ is deduced, the existence of the maximal solution and the minimal solution is discussed and two new iterative algorithms for obtaining these solutions are proposed. These algorithms avoid matrix inversion.

Introduction

In this paper, we will discuss the following nonlinear matrix equation

$$X^s+A^{\ast }{X}^{-q}A=I\text{,}$$

where $A\in {\mathbf{C}}^{m\times n}\text{,}s\text{,}q\in {\mathbf{R}}^{+}$ and $X\in {\mathbf{C}}^{m\times m}$ is an unknown Hermitian positive definite (HPD) matrix to be found.

Eq. (1) often arises in dynamic programming, control theory, stochastic filtering, statistics, and so on, see [1]. The case for $s=q=1$ has been extensively studied by several authors [1], [2], [3], [4], [5], [6]. It was proved in [2] that if Eq. (1) has an HPD solution, then all its Hermitian solutions are positive definite; and moreover, it has the maximal solution $X_L$ and the minimal solution $X_S$ in the sense that $X_S⩽X⩽X_L$ for any HPD solution X. Here $X⩾Y$ means that $X-Y$ is Hermitian positive semidefinite and if a Hermitian matrix Z satisfies $Y⩽Z⩽X$, then we say $Z\in [Y\text{,}X]$.

The other cases have been studied in [7], [8], [9], [10], [11], [12], [13], [14], but the existence of the maximal solution and the minimal solution has not been detailedly discussed.

In this paper, we will discuss the distribution of the HPD solution and the existence of the maximal solution and the minimal solution, and propose two iterative algorithms for finding these solutions. Our algorithms will avoid matrix inversion.

For $A\in {\mathbf{C}}^{n\times n}\text{,}\lambda (A)\text{,}{\lambda }_{\max }(A)$ , ${\lambda }_{\min }(A)$ and $\Vert A\Vert$ denote the eigenvalue, the maximal eigenvalue, the minimal eigenvalue and the spectral norm of A, respectively.

In the remainder of this section, we will give some results for later discussion.

Lemma 1.1 [15]

If $A>B>0$ (or $A⩾B>0$), then $A^{\alpha }>B^{\alpha }$ (or $A^{\alpha }⩾B^{\alpha }>0$) for all $\alpha \in (0\text{,}1]$, and $A^{\alpha }<B^{\alpha }$ (or $0<A^{\alpha }⩾B^{\alpha }$) for all $\alpha \in [-1\text{,}0)$.

Lemma 1.2 [5]

If C and P are Hermitian matrices of the same order with $P>0$, then $\mathit{CPC}+P^{-1}⩾2C$.

Lemma 1.3 [15]

If $0<\alpha ⩽1$, and P and Q are positive definite matrices of the same order with $P\text{,}Q⩾\mathit{bI}>0$, then $\Vert P^{\alpha }-Q^{\alpha }\Vert ⩽\alpha b^{\alpha -1}\Vert P-Q\Vert$.

Lemma 1.4 [16]

If $0<\alpha$, and P and Q are positive definite matrices of the same order with $P\text{,}Q⩾\mathit{bI}>0$, then $\Vert P^{-\alpha }-Q^{-\alpha }\Vert ⩽\alpha b^{-\alpha -1}\Vert P-Q\Vert$.