Hermitian Positive Definite Solution of the Matrix Equation

We consider the Hermitian positive definite solution of the nonlinear matrix equation . Some new sufficient conditions and necessary conditions for the existence of Hermitian positive definite solutions are derived. An iterative method is proposed to compute the Hermitian positive definite solution. In the end, an example is used to illustrate the correctness and application of our results.


Introduction
In this paper we consider the Hermitian positive definite solution of the nonlinear matrix equation where  1 ,  2 , . . .,   are  ×  complex matrices;  and  are  ×  Hermitian positive definite matrices.Here,  *  denotes the conjugate transpose of the matrix   .The nonlinear matrix equation (1) plays an important role in linear optimal and robust control.For instance, the solvability of the discrete-time linear quadratic optimal control problem min with  *  =   ,  *  =   depends on the solvability of (1) in some special cases [1][2][3][4].Due to the important applications in system and control theory, in the past decades, (1) with  = 1 has been extensively studied, and the research results mainly concentrated on the following: (a) sufficient conditions and necessary conditions for the existence of an Hermitian solution [5][6][7][8][9]; (b) numerical methods for computing the Hermitian solution [4,[10][11][12][13]; (c) properties of the Hermitian solution [14,15]; (d) perturbation analysis for the discrete algebraic Riccati equation [16][17][18].
Nonetheless, (1) with  > 1 has not been studied as far as we know.
In this paper we study the generalized nonlinear matrix equation (1).Firstly, we transform (1) into an equivalent nonlinear matrix equation.By Sherman-Woodbury-Morrison formula [24, Page 50], we have International Journal of Computational Mathematics then Set then That is, Therefore, the nonlinear matrix equation ( 1) can be equivalently rewritten as (7).So we first investigate the Hermitian positive definite solution of (7) in Section 2 and then derive some new results on the nonlinear matrix equation (1) by using the matrix transformations (5) in Section 3. Finally, we use an example to illustrate the correctness and application of the results of Section 3. Throughout this paper, we write  > 0 ( ≥ 0) if the matrix  is Hermitian positive definite (semidefinite).If − is Hermitian positive definite (semidefinite), then we write  >  ( ≥ ).If an Hermitian positive definite matrix  satisfies  <  ≤ , we denote by  ∈ (, ].We use  1 () ≥  2 () ≥ ⋅ ⋅ ⋅ ≥   () to denote all eigenvalues (each repeated as many times as its algebraic multiplicity) of an  ×  Hermitian matrix .The symbol ‖‖ denotes the spectral norm of the matrix .(7) In this section, we first give some sufficient conditions and necessary conditions for the existence of an Hermitian positive definite solution of (7), and then we construct an iterative method to compute the Hermitian positive definite solution.We begin with some lemmas.

Hermitian Positive Definite Solution of
Lemma 1 (see [25]).If  ≥  > 0, then  −1 ≤  −1 .Lemma 2. If (7) has an Hermitian positive definite solution , then Proof.Since (7) has an Hermitian positive definite solution , that is, then And from Lemma 1 it follows that which leads to that is, Lemma 3. Suppose that the matrices   ,  2 , . . .,   are nonsingular; if (7) has an Hermitian positive definite solution , then Proof.Since (7) has an Hermitian positive definite solution , that is, then which implies By (17) and Lemma 1 and noting that the matrices   ,  2 , . . .,   are nonsingular, then we have which leads to that is, By ( 16) and ( 20), we have Theorem 4. Let If then (7) has an Hermitian positive definite solution X and X ∈ [, ], where Proof.Since and noting that then we have Therefore, the quadratic equation has a positive root , and the quadratic equation has a positive root , where  and  are defined by (24).Now we consider the map Obviously, Ω is a convex, closed, and bounded set and the map  is continuous on Ω.For arbitrary  ∈ Ω, since  and  are positive roots of (28) and (29), respectively, then we have that is, () ∈ Ω, which implies that (Ω) ⊆ Ω.By Brouwer's fixed point theorem, the map  has a fixed point X ∈ Ω, which is the positive definite solution of (7).The theorem is proved.
, then (7) has an Hermitian positive definite solution.Furthermore, if then (7) has a unique Hermitian positive definite solution.
Proof.Consider the map Obviously,  is a convex, closed, and bounded set and the map  is continuous on .
for all  ∈ , then we have (36) that is, Hence, () ⊆ .By Brouwer's fixed point theorem, the map  has a fixed point in , which is the Hermitian positive definite solution of (7).
For arbitrary ,  ∈ , we have Nextly, we give an iterative method to compute the Hermitian positive definite solution of (7).Theorem 6.If there is a real number  satisfying then the matrix sequence {  } converges the Hermitian positive definite solution of (7), where the sequence {  } is generated by the following iterative method: Proof.Consider the iterative method (40).From (39) it follows that Since  1 ≤  0 , then which implies that Therefore, Assume that when  = , we have   ≤  −1 ; then which implies that By mathematical induction, we obtain that the matrix sequence {  } is monotone decreasing.Now we begin to use mathematical induction to show that the matrix sequence {  } is bounded below by some Hermitian positive definite solution of (7).
When  = 0, by (39) we have Assume that when  = , we have   ≥ ; then ( −1 +   ) −1 ≤ ( −1 + ) −1 , and for  =  + 1 we have Therefore, the matrix sequence {  } is a monotonic decreasing sequence and bounded below by some Hermitian positive definite solution , and then it converges to an Hermitian positive definite solution  of (1).The theorem is proved.
Remark 9.The upper bounds of Hermitian solution for the discrete algebraic Riccati equation have been extensively studied in the past three decades (see [14] and references therein).Compared with previous works, one gives a new upper bound of Hermitian solution but also gives a lower bound in Theorem 8.

Numerical Experiments
In this section, we give an example to demonstrate the correctness and application of our results.(59)