# Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix EquationX+∑i=1mAiTXδiAi=I

Authors: Naglaa M. El-Shazly
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

In this paper necessary and sufficient conditions for the matrix equationX+∑i=1mAiTXδiAi=I to have a positive definite solution X are derived, where -1<δi<0, I is an n×n identity matrix, Ai are n×n nonsingular real matrices, and m is an odd positive integer. These conditions are used to propose some properties on the matrices Ai, i=1,2,…,m. Moreover, relations between the solution X and the matrices Ai are derived.


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## Body

## 1. Introduction

Consider the nonlinear matrix equation(1)X+∑i=1mAiTXδiAi=I,-1<δi<0,
where I is an n×n identity matrix, Ai are n×n nonsingular real matrices, and m is an odd positive integer (Ai* stands for the conjugate transpose of the matrix Ai). 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 Section2, 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 Ai are given.


## 2. Preliminaries

In this section, we present some notations, lemmas, and theorems that will be needed to develop this paper. We denote byr(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.
LetE and F be two arbitrary compatible matrices. Then,
(2)r(ETF-FTE)≤r(ETE+FTF).The proof of this lemma is given in [14, Lemma 6].Theorem 2 (Cholesky Decomposition).
IfG is an n×n symmetric positive definite matrix, then there exists a triangular n×n matrix M with positive diagonal entries such that G=MTM.For proof, see [15, page 141].Theorem 3 (C-S decomposition [15, page 77] and [16, page 37]).
IfQ is an orthogonal matrix where Q=[Q11Q12Q21Q22], then there exist orthogonal matrices U1, U2, V1, V2 and diagonal matrices Γ≥0, Σ≥0  such that
(3)Q=[U100U2][Γ-ΣΣΓ][V100V2],
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 Ai, i=1,2,…,m, admit the following factorization:
(4)∑i=1mAi=∑i=1m(WTW)-δi/2Zi,
where  W is a nonsingular square matrix and the columns of
(5)(WZ1Z2⋮Zm)
are orthonormal. In this case, we take X=WTW as a solution of the matrix equation (1).Proof.
If (1) has a solution  X, then one can write X as  X=WTW for some nonsingular matrix W. In particular, W can be chosen to be triangular using Theorem 2; thus, we have
(6)X+∑i=1mAiTXδiAi=WTW+∑i=1mAiT(WTW)δiAi=WTW+∑i=1mAiT(WTW)δi/2(WTW)δi/2Ai.
Then (1) can be rewritten as
(7)WTW+∑i=1m((WTW)δi/2Ai)T((WTW)δi/2Ai)=I,
or equivalently
(8)[WT,∑i=1m((WTW)δi/2Ai)T][W∑i=1m((WTW)δi/2Ai)]=I.
Let
(9)∑i=1m((WTW)δi/2Ai)=∑i=1mZi.
Then
(10)∑i=1mAi=∑i=1m(WTW)-δi/2Zi,
and  (8) means that the columns of
(11)(WZ1Z2⋮Zm)
are orthonormal.
Conversely, suppose thatAi, i=1,2,…,m have the decomposition (4). Set X=WTW. Then
(12)X+∑i=1mAiTXδiAi=WTW+∑i=1m((WTW)-δi/2Zi)T×(WTW)δi((WTW)-δi/2Zi)=WTW+∑i=1m(ZiT(WTW)-δi/2)×(WTW)δi/2(WTW)δi/2×((WTW)-δi/2Zi)=WTW+∑i=1mZiTZi=I.
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
(13)∑i=1mAi=∑(iodd)m(PTΓ2P)-δi/2Ui+1ΣP+∑(ieven)m-1(PTΓ2P)-δi/2Ui+1ΓP.
In this case, X=PTΓ2P is a solution of (1).Proof.
Suppose that the matrix equation (1) has a solution. Then from Theorem 4,   Ai, i=1,2,…,m, have the form
(14)∑i=1mAi=∑i=1m(WTW)-δi/2Zi.
Since(15)(WZ1Z2⋮Zm)
is column-orthonormal, it can be extended to an orthogonal matrix(16)(WUZ1V1Z2V2⋮⋮ZmVm),
and applying Theorem3, there exist orthogonal matrices U1,U2,…,Um+1,P,Q and diagonal matrices Γ>0 and Σ≥0 such that
(17)(WUZ1V1Z2V2⋮⋮ZmVm)=(U100U2⋮⋮Um00Um+1)(Γ-ΣΣΓ)(P00Q),
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.
ThusW=U1ΓP, and
(18)∑i=1mZi=∑i=1,3mUi+1ΣP+∑i=2,4m-1Ui+1ΓP.
Then (4) can be rewritten as follows:
(19)∑i=1mAi=∑i=1m(WTW)-δi/2Zi=∑i=1,3m(PTΓ2P)-δi/2Ui+1ΣP+∑i=2,4m-1(PTΓ2P)-δi/2Ui+1ΓP.
Conversely, suppose thatAi, i=1,2,…,m have the decomposition (13). Let X=PTΓ2P. Then
(20)X+∑i=1mAiTXδiAi=(U1ΓP)T(U1ΓP)+∑i=1,3m((PTΓ2P)-δi/2Ui+1ΣP)T(PTΓ2P)δi×((PTΓ2P)-δi/2Ui+1ΣP)+∑i=2,4m-1((PTΓ2P)-δi/2Ui+1ΓP)T(PTΓ2P)δi×((PTΓ2P)-δi/2Ui+1ΓP)=PTΓ2P+((PTΓ2P)-δ1/2U2ΣP)T×(PTΓ2P)δ1((PTΓ2P)-δ1/2U2ΣP)+((PTΓ2P)-δ3/2U4ΣP)T(PTΓ2P)δ3×((PTΓ2P)-δ3/2U4ΣP)+⋯+((PTΓ2P)-δm/2Um+1ΣP)T×(PTΓ2P)δm((PTΓ2P)-δm/2Um+1ΣP)+((PTΓ2P)-δ2/2U3ΓP)T(PTΓ2P)δ2×((PTΓ2P)-δ2/2U3ΓP)+⋯+((PTΓ2P)-δm-1/2UmΓP)T×(PTΓ2P)δm-1((PTΓ2P)-δm-1/2UmΓP)=PTΓ2P+(PTΣU2T)(PTΓ2P)-δ1/2×(PTΓ2P)δ1(PTΓ2P)-δ1/2U2ΣP+(PTΣU4T)(PTΓ2P)-δ3/2×(PTΓ2P)δ3(PTΓ2P)-δ3/2U4ΣP+…+(PTΣUm+1T)(PTΓ2P)-δm/2×(PTΓ2P)δm(PTΓ2P)-δm/2Um+1ΣP+(PTΓU3T)(PTΓ2P)-δ2/2×(PTΓ2P)δ2(PTΓ2P)-δ2/2U3ΓP+(PTΓUmT)(PTΓ2P)-δm-1/2×(PTΓ2P)δm-1(PTΓ2P)-δm-1/2UmΓP=PTΓ2P+PTΣU2TU2ΣP+PTΣU4TU4ΣP+⋯+PTΣUm+1TUm+1ΣP+PTΓU3TU3ΓP+⋯+PTΓUmTUmΓP=PTΓ2P+PTΣ2P+PTΣ2P+⋯+PTΣ2P+PTΓ2P+⋯+PTΓ2P=PT(m+12(Γ2+Σ2))P=PTP=I.
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  Ai.Theorem 6.
If the matrix equation (1) has a positive definite solution  X, then ∥∑i=1mAi∥<m.Proof.
Suppose that (1) has a solution. Then by Theorem 5, ∑i=1mAi has the decomposition (13).
Then(21)∥∑i=1mAi∥=∥∑i=1,3m(PTΓ2P)-δi/2Ui+1ΣP+∑i=2,4m-1(PTΓ2P)-δi/2Ui+1ΓP∥≤∥∑i=1,3m(PTΓ2P)-δi/2Ui+1ΣP∥+∥∑i=2,4m-1(PTΓ2P)-δi/2Ui+1ΓP∥≤∑i=1,3m∥(PTΓ2P)-δi/2∥∥Ui+1∥∥Σ∥∥P∥+∑i=2,4m-1∥(PTΓ2P)-δi/2∥∥Ui+1∥∥Γ∥∥P∥.
Since each Ui+1, i=1,2,…,m, is orthogonal, also P is orthogonal. So
(22)∥∑i=1mAi∥≤∑i=1,3m∥(PTΓ2P)-δi/2∥∥Σ∥+∑i=2,4m-1∥(PTΓ2P)-δi/2∥∥Γ∥≤∑i=1,3m∥PTΓ2P∥-δi/2∥Σ∥+∑i=2,4m-1∥PTΓ2P∥-δi/2∥Γ∥=∑i=1,3m∥Γ2∥-δi/2∥Σ∥+∑i=2,4m-1∥Γ2∥-δi/2∥Γ∥≤∑i=1,3m∥Γ∥-δi∥Σ∥+∑i=2,4m-1∥Γ∥-δi∥Γ∥.
Since we choose -1<δi<0, then we have ∥Γ∥-δi≤∥Γ∥<1, and therefore
(23)∑i=1,3m∥Γ∥-δi≤(m+12),∑i=2,4m-1∥Γ∥-δi≤(m-12).
Thus,
(24)∥∑i=1mAi∥≤(m+12)(2m+1)+(m-12)(2m+1)=m2m+1<m.
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)
∑i=1mX-δi>∑i=1mAiAiT,(ii)
I-∑i=1mAiTXδi+1Ai-∑i=1mX(δi+1)/2AiAiTX(δi+1)/2>0,(iii)
r(∑i=1mAi)≤(1/2)2/(m+1)((3m-1)/(m+1)),(iv)
r(∑i=1mX(1+δi)/2Ai-∑i=1mAiTX(1+δi)/2)≤1,(v)
r(∑i=1mX(1+δi)/2Ai+∑i=1mAiTX(1+δi)/2)≤1.Proof.
(i) By Theorem4, we have X=WTW and ∑i=1mAi=∑i=1m(WTW)-δi/2Zi.  Then
(25)∑i=1mX-δi-∑i=1mAiAiT=∑i=1m(WTW)-δi-∑i=1m(WTW)-δi/2ZiZiT(WTW)-δi/2=∑i=1m(WTW)-δi/2(WTW)-δi/2-∑i=1m(WTW)-δi/2ZiZiT(WTW)-δi/2=∑i=1m(WTW)-δi/2[I-∑i=1mZiZiT](WTW)-δi/2>0,
sinceλ(ZiZiT)=λ(ZiTZi), i=1,2,…,m, and I-∑i=1mZiTZi=WTW>0, and, hence, I-∑i=1mZiZiT>0.  Thus, part (i) is proved.
(ii) ReplacingI by WTW+∑i=1mZiTZi, we get
(26)I-∑i=1mAiTXδi+1Ai-∑i=1mX(δi+1)/2AiAiTX(δi+1)/2=WTW+∑i=1mZiTZi-∑i=1mAiTXδi+1Ai-∑i=1mX(δi+1)/2AiAiTX(δi+1)/2=WTW+∑i=1mZiTZi-∑i=1m((WTW)-δi/2Zi)T×(WTW)δi+1((WTW)-δi/2Zi)-∑i=1m(WTW)(δi+1)/2((WTW)-δi/2Zi)×((WTW)-δi/2Zi)T(WTW)(δi+1)/2=WTW+∑i=1mZiTZi-∑i=1mZiT(WTW)-δi/2×(WTW)δi+1(WTW)-δi/2Zi-∑i=1m(WTW)(δi+1)/2((WTW)-δi/2Zi)×(ZiT(WTW)-δi/2)(WTW)(δi+1)/2=(WTW)1/2(WTW)1/2+∑i=1mZiTZi-∑i=1mZiT(WTW)Zi-(WTW)1/2∑i=1mZiZiT(WTW)1/2=(WTW)1/2(I-∑i=1mZiZiT)×(WTW)1/2+∑i=1mZiT(I-WTW)Zi>0.
This completes the proof of (ii).
(iii) Using Theorem6, we get
(27)λ(∑i=1mAi)=λ(∑i=1,3m(PTΓ2P)-δi/2Ui+1ΣP+∑i=2,4m-1(PTΓ2P)-δi/2Ui+1ΓP),r(∑i=1mAi)=maxi|λ(∑i=1,3m(PTΓ2P)-δi/2Ui+1ΣP+∑i=2,4m-1(PTΓ2P)-δi/2Ui+1ΓP)|≤∥(∑i=1,3m(PTΓ2P)-δi/2Ui+1ΣP+∑i=2,4m-1(PTΓ2P)-δi/2Ui+1ΓP)∥≤∥∑i=1,3m(PTΓ2P)-δi/2Ui+1ΣP∥+∥∑i=2,4m-1(PTΓ2P)-δi/2Ui+1ΓP∥=∑i=1,3m∥(PTΓ2P)-δi/2Σ∥+∑i=2,4m-1∥(PTΓ2P)-δi/2Γ∥.
Since we choose-1<δi<0, then we have 0<(-δi)/2<1/2<1, and therefore (PTΓ2P)-δi/2≤(PTΓ2P), i=1,2,…,m; then
(28)r(∑i=1mAi)≤m+12∥(PTΓ2P)Σ∥+m-12∥(PTΓ2P)Γ∥=m+12∥PTΣΓ2P∥+m-12∥PTΓ3P∥=m+12∥ΣΓ2∥+m-12∥Γ3∥≤m+12∥ΣΓΓ∥+m-12∥Γ∥3≤m+12∥ΣΓ∥∥Γ∥+m-12∥Γ∥3.
LetΣ=diag(σj), Γ=diag(γj). Then ((m+1)/2)(σj2+γj2)=1. Thus
(29)r(∑i=1mAi)≤m+122m+1∥ΣΓ∥+m-12(2m+1)3=m+122m+1maxj{σjγj}+m-12(2m+1)3≤m+122m+1maxjσj2+γj22+m-12(2m+1)3=122m+1(3m-1m+1).
(iv) Appling Lemma1, we get
(30)r(∑i=1mX(1+δi)/2Ai-∑i=1mAiTX(1+δi)/2)=r(∑i=1m(WTW)(1+δi)/2(WTW)-δi/2Zi-∑i=1mZiT(WTW)-δi/2(WTW)(1+δi)/2)=r(∑i=1m(WTW)1/2Zi-∑i=1mZiT(WTW)1/2)≤r((WTW)+∑i=1mZiTZi)=r(I)=1.
(v) Consider(31)I±(∑i=1mX(1+δi)/2Ai+∑i=1mAiTX(1+δi)/2)=WTW+∑i=1mZiTZi±(∑i=1m(WTW)1/2Zi+∑i=1mZiT(WTW)1/2)=((WTW)1/2±∑i=1mZi)T((WTW)1/2±∑i=1mZi)≥0.
Therefore, r(∑i=1mX(1+δi)/2Ai+∑i=1mAiTX(1+δi)/2)≤1.


## 5. Conclusion

In this paper, both necessary and sufficient conditions for the nonlinear matrix equationX+∑i=1mAiTXδiAi=I to have a positive definite solution X are given, where -1<δi<0, Ai 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 Ai, i=1,2,…,m, are presented.



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