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

Authors: Wenling Zhao; Hongkui Li; Xueting Liu; Fuyi Xu
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We study the Hermitian positive definite solutions of the nonlinear matrix equationX+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.


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## Body

## 1. Introduction

In this paper, we conside the Hermitian positive definite solutions of the matrix equation:(1.1)X+A*X-2A=I,
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]:
(1.2)Mx=f,
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 Xs and special Xl 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∥=maxiλi, where the λi 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, if0<A≤B, then ∥A∥≤∥B∥.Lemma 2.2.
IfM 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]).
IfX is a positive definite solution of (1.1), then
(2.1)I<X<I+A*A.Lemma 2.4 (see [13, Theorem  4]).
If (1.1) has a solution, then
(2.2)∥A∥<1.
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
(3.1)A=U*Λ2ZU,
where Λ=diag(λ1,λ2,…,λs), λi are the eigenvalues of X,  i=1,2,…,n,  X=U*Λ2U, and the columns of
(3.2)(ΛZ)
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
(3.3)Λ1+C*Λ1-2C=I,
where C=UAU* is a nonsingular Hermitian matrix, Λ1=diag(λ1,λ2,…,λs), λi 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,…,λn),λi(i=1,2,…,n) are the eigenvalues of X. Let Λ1=Λ2, then Λ=diag(λ1,λ2,…,λs), we rewrite the matrix equation (1.1) as
(3.4)U*Λ2U+A*(U*Λ2U)-2A=I,
so
(3.5)U*Λ2U+A*U*Λ-4UA=I.
Because U is a unitary matrix, hence
(3.6)Λ2+(U*)-1A*U*Λ-4UAU-1=(U*)-1IU-1=I.
Thus
(3.7)(ΛΛ-2UAU*)*(ΛΛ-2UAU*)=I.
Let Z=Λ-2UAU*, then A=U*Λ2ZU and by (3.3), it is easy to see that the columns of
(3.8)(ΛZ)
are orthonormal.
Conversely, suppose thatA has the decomposition (3.1) and X=U*Λ2U, where U is a unitary matrix. Since the columns of
(3.9)(ΛZ)
are orthonormal, so
(3.10)(ΛΛ-2UAU*)*(ΛΛ-2UAU*)=I,
that is,
(3.11)Λ2+(U*)-1A*U*Λ-4UAU-1=(U*)-1IU-1=I.
Then, we have(3.12)X+A*X-2A=U*Λ2U+A*(U*ΛU)-4A=U*ΛU+(U*Λ2ZU)*U*Λ-4UU*Λ2ZU=I.
HenceX is a Hermitian positive definite solution of (1.1).
The proof of Theorem3.1 is complete.Theorem 3.2.
If the matrix equation (1.1) has a Hermitian positive definite solution X, then
(3.13)∥A∥1/2<λ¯<1,λ̲<1-∥A∥,
where λ¯=max{λi,i=1,2,…,n},λ̲=min{λi,i=1,2,…,n},λi,i=1,2,…,n are the eigenvalues of X.Proof.
By Theorem3.1, if Λ1 is a Hermitian positive definite solution of (3.3), then we have
(3.14)Λ1>0,C*Λ1-2C>0,
thus
(3.15)C*Λ1-2C=I-Λ1>0,
hence I-Λ1>0, that is,
(3.16)0<λi<1,
where λi,i=1,2,…,n.
Without loss of generality, suppose0<λ1<λ2<⋯<λn<1, then λ¯=λn<1. By the proof of Theorem 3.1, we have
(3.17)(Λ1-1UA)*(Λ1-1UA)=U*(I-Λ1)U.
Let ∑=diag(σ1,σ2,…,σn)≥0 such that Λ1+∑2=I, where σ1≥σ2≥⋯≥σn. Since λ¯<1, then ∑=diag(σ1,σ2,…,σn)>0, 1>σ1≥σ2≥⋯≥σn>0, and σ1=1-λ1, that is,
(3.18)σ1=1-λ̲.
Now let(3.19)B=Λ1-1UAU*=(b1,b2,…,bn),
where bi is n-column vector, then B*B=∑, so B is column orthogonal and bi*bi=σi2,i=1,2,…,n. Let vi=bi/σi,i=1,2,…,n, and with CS factorization theorem, extend them to form an orthogonal basis of Cn. Thus V=(v1,v2,,vn) is a unitary matrix such that B=V∑, and hence
(3.20)A=U*Λ1V∑U.
So
(3.21)∥A∥=∥U*Λ1V∑U∥=∥Λ1V∑∥≤∥Λ1∥∥∑∥=λ¯2(1-λ̲),
thus we have
(3.22)∥A∥≤λ¯2,
that is,
(3.23)∥A∥1/2≤λ¯,∥A∥≤1-λ̲,
that is,
(3.24)λ̲≤1-∥A∥.
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{λi2,i=1,2,…,n},β=max{1-λi,i=1,2,…,n},λ̲=min{λi,i=1,2,…,n}, where λi,i=1,2,…,n are the eigenvalues of X, then(1)
A*A<βΛ2<βαI;(2)
λ̲I<Λ<I-αA*A.Proof.
By Theorem3.1, C=UAU*, where U is a unitary matrix, hence A*A=UC*CU*.
Letα=max{λi2,i=1,2,…,n},β=max{1-λi,i=1,2,…,n},λ̲=min{λi,i=1,2,…,n}, where λi,i=1,2,…,n be the eigenvalues of X.
(1) By Theorem3.1, Λ1 is a Hermitian positive definite solution of (3.3), thus
(3.25)Λ<αI,C*Λ1-2C=I-Λ1<I,
hence
(3.26)C*Λ1-2C=(I-Λ1)<βI,Λ12<αI<I.
So(3.27)A*A=U*C*CU<βU*Λ12U<αβI.
(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
(3.28)C*Λ1-2C>αC*C,
so
(3.29)αA*A=UαC*CU*<I-Λ1,
thus we have
(3.30)λ̲I<Λ1<I-αA*A.
The proof of Theorem3.3 is complete.Theorem 3.4.
SupposeA and B are two positive definite Hermitian matrices, and A>B, then there exist a nonsingular matrices T which satisfied(1)
(3.31)T*AT=I,T*BT=Λ,
where Λ=diag(λ1,λ2,…,λn), λi are the eigenvalues of A-1B,i=1,2,…,n;(2)
0<λi<1, and the columns of
(3.32)(Λ1/2(A-B)1/2T)
are orthonormal, i=1,2,…,n.Proof.
(1) Since the matrixA is positive definite Hermitian, then there exists a unique positive definite Hermitian matrix W such that A=W2. We note
(3.33)F=(W-1)*BW-1=WBW-1.
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,…,λn), λi are the eigenvalues of F. Hence we have the following equation
(3.34)U*(W-1)*BW-1U=Λ.
Let T=W-1U, then T is nonsingular. Since
(3.35)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),
so the eigenvalue of A-1B is equal to the eigenvalue of F.
(2) If (3.31) is hold, then we have
(3.36)T*(A-B)T=I-Λ.
Since A>B,  T is nonsingular, so T*(A-B)T is positive definite Hermitian.
ThusI-Λ>0, that is, 0<λi<1,i=1,2,…,n. It is easy to see that the columns of
(3.37)(Λ1/2(A-B)1/2T)
are orthonormal.
The proof of Theorem3.4 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:
(3.38)A=TΛ-1/2Z,
where Λ=diag(λ1,λ2,…,λn),λi,i=1,2,…,n are the eigenvalues of X-3,T*XT=I,T*X-2T=Λ and the columns of
(3.39)(T-1Z)
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
(3.40)T*XT=I,T*X-2T=Λ,
hence
(3.41)X=(T-1)*T-1,X-2=(T-1)*ΛT-1.
So (1.1) is equivalent to
(3.42)(T-1)*T-1+A*(T-1)*ΛT-1A=I,
or equivalently
(3.43)(T-1Λ1/2T-1A)*(T-1Λ1/2T-1A)=I.
Let  Λ1/2T-1A=Z, then A=TΛ-1/2Z and (3.43) means that the columns of
(3.44)(T-1Λ1/2T-1A)
are orthonormal.
Conversely, suppose thatA has the decomposition A=TΛ-1/2Z and X=(T-1)*T-1, where U is a unitary matrix. Since the columns of
(3.45)(T-1Z)
are orthonormal, so
(3.46)(T-1Λ1/2T-1A)*(T-1Λ1/2T-1A)=I.
Hence
(3.47)(T-1)*T-1+A*(T-1)*ΛT-1A=I.
Then we have
(3.48)X+A*X-2A=(T-1)*T-1+(T-1)*ΛT-1=I.
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:
(3.49)A=TZ,
where Λ=diag(λ1,λ2,…,λn),λi,i=1,2,…,n are the eigenvalues of X3,T*XT=Λ,T*X-2T=I and the columns of
(3.50)(Λ1/2T-1Z)
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
(3.51)T*XT=Λ,T*X-2T=I,
hence
(3.52)X=(T-1)*ΛT-1,X-2=(T-1)*T-1.
So (1.1) is equivalent to
(3.53)(T-1)*ΛT-1+A*(T-1)*T-1A=I,
or equivalently
(3.54)(Λ1/2T-1T-1A)*(Λ1/2T-1T-1A)=I.
Let T-1A=Z, then A=TZ, and (3.54) means that the columns of
(3.55)(Λ1/2T-1T-1A)
are orthonormal.
Conversely, suppose thatA has the decomposition A=TZ and X=(T-1)*ΛT-1, where U is a unitary matrix. Since the columns of
(3.56)(Λ1/2T-1Z)
are orthonormal, then
(3.57)(Λ1/2T-1T-1A)*(Λ1/2T-1T-1A)=I.
Thus
(3.58)(T-1)*ΛT-1+A*(T-1)*T-1A=I.
Then we have
(3.59)X+A*X-2A=(T-1)*ΛT-1+(T-1)*T-1=I.
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,…,λn),λi,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
(3.60)T*XT=I,T*A*X-2AT=Λ1,
hence
(3.61)X=(T-1)*T-1,A*X-2A=(T-1)*Λ1T-1.
So (1.1) is equivalent to
(3.62)(T-1)*T-1+(T-1)*Λ1T-1=I,
that is,
(3.63)(T-1)*(I+Λ1)T-1=I.
Let Λ=Λ1+I, then (3.63) is equivalent to
(3.64)(T-1)*Λ2T-1=I.
So
(3.65)(ΛT-1)*(ΛT-1)=I.
Thus ΛT is unitary.
The proof of Theorem3.7 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,…,λn),λi,i=1,2,…,n are the eigenvalues of A-1X2(A-1)*X.Remark 3.9.
By Theorems3.7 and 3.8, the solving of  (1.1) is transformed to the solving of
(3.66)T*T=Λ,
where Λ=diag(λ1-1,λ2-1,…,λn)-1,λi,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 eigenvalueλi of A-1X2(A-1)*X or X-1A*X-2A satisfied
(3.67)λi>1.



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