Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 537520, 7 pages
http://dx.doi.org/10.1155/2013/537520
Research Article
Necessary and Sufficient Conditions for the Existence of
a Positive Definite Solution for the Matrix Equation
𝑇 𝛿𝑖
𝐴
𝑋 + ∑𝑚
𝑖=1 𝑖 𝑋 𝐴 𝑖 = 𝐼
Naglaa M. El-Shazly
Department of Mathematics, Facultyof Science, Menoufia University, Shebin El-Koom, Egypt
Correspondence should be addressed to Naglaa M. El-Shazly; naglaamoh1@yahoo.com
Received 19 December 2012; Revised 26 March 2013; Accepted 14 May 2013
Academic Editor: Theodore E. Simos
Copyright © 2013 Naglaa M. El-Shazly. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
𝑚
In this paper necessary and sufficient conditions for the matrix equation 𝑋 + ∑𝑖=1 𝐴𝑇𝑖 𝑋𝛿𝑖 𝐴 𝑖 = 𝐼 to have a positive definite solution
𝑋 are derived, where −1 < 𝛿𝑖 < 0, 𝐼 is an 𝑛 × 𝑛 identity matrix, 𝐴 𝑖 are 𝑛 × 𝑛 nonsingular real matrices, and 𝑚 is an odd positive
integer. These conditions are used to propose some properties on the matrices 𝐴 𝑖 , 𝑖 = 1, 2, . . . , 𝑚. Moreover, relations between the
solution 𝑋 and the matrices 𝐴 𝑖 are derived.
denote by 𝑟(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).
1. Introduction
Consider the nonlinear matrix equation
𝑚
𝑋 + ∑ 𝐴𝑇𝑖 𝑋𝛿𝑖 𝐴 𝑖 = 𝐼,
𝑖=1
−1 < 𝛿𝑖 < 0,
(1)
where 𝐼 is an 𝑛 × 𝑛 identity matrix, 𝐴 𝑖 are 𝑛 × 𝑛 nonsingular
real matrices, and 𝑚 is an odd positive integer (𝐴∗𝑖 stands for
the conjugate transpose of the matrix 𝐴 𝑖 ). 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 Section 2, 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 𝐴 𝑖 are given.
2. Preliminaries
In this section, we present some notations, lemmas, and
theorems that will be needed to develop this paper. We
Lemma 1. Let 𝐸 and 𝐹 be two arbitrary compatible matrices.
Then,
𝑟 (𝐸𝑇 𝐹 − 𝐹𝑇 𝐸) ≤ 𝑟 (𝐸𝑇 𝐸 + 𝐹𝑇 𝐹) .
(2)
The proof of this lemma is given in [14, Lemma 6].
Theorem 2 (Cholesky Decomposition). If 𝐺 is an 𝑛 × 𝑛 symmetric positive definite matrix, then there exists a triangular
𝑛 × 𝑛 matrix 𝑀 with positive diagonal entries such that 𝐺 =
𝑀𝑇 𝑀.
For proof, see [15, page 141].
Theorem 3 (C-S decomposition [15, page 77] and [16, page
11 𝑄12
37]). If 𝑄 is an orthogonal matrix where 𝑄 = [ 𝑄
𝑄21 𝑄22 ], then
there exist orthogonal matrices 𝑈1 , 𝑈2 , 𝑉1 , 𝑉2 and diagonal
matrices Γ ≥ 0, Σ ≥ 0 such that
Γ −Σ 𝑉1 0
𝑈 0
][
],
𝑄=[ 1
][
0 𝑈2 Σ Γ
0 𝑉2
where Γ2 + Σ2 = 𝐼.
(3)
2
Journal of Applied Mathematics
3. Necessary and Sufficient Conditions
and (8) means that the columns of
𝑊
𝑍1
( 𝑍2 )
..
.
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 𝐴 𝑖 , 𝑖 = 1, 2, . . . , 𝑚,
admit the following factorization:
𝑚
𝑚
𝑖=1
𝑖=1
∑ 𝐴 𝑖 = ∑ (𝑊𝑇 𝑊)
−𝛿𝑖 /2
𝑍𝑖 ,
(4)
(11)
𝑍𝑚
are orthonormal.
Conversely, suppose that 𝐴 𝑖 , 𝑖 = 1, 2, . . . , 𝑚 have the decomposition (4). Set 𝑋 = 𝑊𝑇 𝑊. Then
𝑚
where 𝑊 is a nonsingular square matrix and the columns of
𝑋 + ∑ 𝐴𝑇𝑖 𝑋𝛿𝑖 𝐴 𝑖
𝑖=1
𝑚
𝑊
𝑍1
( 𝑍2 )
..
.
= 𝑊𝑇 𝑊 + ∑ ((𝑊𝑇 𝑊)
−𝛿𝑖 /2
𝑖=1
(5)
𝑍𝑖 )
𝑇
𝛿𝑖
−𝛿𝑖 /2
× (𝑊𝑇 𝑊) ((𝑊𝑇 𝑊)
𝑍𝑚
𝑚
= 𝑊𝑇 𝑊 + ∑ (𝑍𝑖𝑇 (𝑊𝑇 𝑊)
are orthonormal. In this case, we take 𝑋 = 𝑊𝑇 𝑊 as a solution
of the matrix equation (1).
−𝛿𝑖 /2
𝑍𝑖 )
(12)
)
𝑖=1
𝛿𝑖 /2
× (𝑊𝑇 𝑊)
Proof. If (1) has a solution 𝑋, then one can write 𝑋 as 𝑋 =
𝑊𝑇 𝑊 for some nonsingular matrix 𝑊. In particular, 𝑊 can
be chosen to be triangular using Theorem 2; thus, we have
𝛿𝑖 /2
(𝑊𝑇 𝑊)
−𝛿𝑖 /2
× ((𝑊𝑇 𝑊)
𝑚
𝑍𝑖 )
𝑚
𝑋 + ∑ 𝐴𝑇𝑖 𝑋𝛿𝑖 𝐴 𝑖
= 𝑊𝑇 𝑊 + ∑ 𝑍𝑖𝑇 𝑍𝑖 = 𝐼.
𝑖=1
𝑖=1
𝑚
𝛿𝑖
= 𝑊𝑇 𝑊 + ∑ 𝐴𝑇𝑖 (𝑊𝑇 𝑊) 𝐴 𝑖
(6)
𝑖=1
𝑚
= 𝑊𝑇 𝑊 + ∑ 𝐴𝑇𝑖 (𝑊𝑇 𝑊)
𝛿𝑖 /2
(𝑊𝑇 𝑊)
𝑖=1
𝛿𝑖 /2
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
𝐴 𝑖.
Then (1) can be rewritten as
𝑚
𝛿𝑖 /2
𝑊𝑇 𝑊 + ∑ ((𝑊𝑇 𝑊)
𝑖=1
𝑚
𝑚
𝑖=1
(𝑖 odd)
−𝛿𝑖 /2
∑ 𝐴 𝑖 = ∑ (𝑃𝑇 Γ2 𝑃)
𝑇
𝐴 𝑖 ) ((𝑊𝑇 𝑊)
𝛿𝑖 /2
𝐴 𝑖 ) = 𝐼,
(7)
𝑈𝑖+1 Σ𝑃
(13)
𝑚−1
−𝛿𝑖 /2
𝑇 2
+ ∑ (𝑃 Γ 𝑃)
or equivalently
(𝑖 even)
𝑚
𝛿𝑖 /2
[𝑊𝑇 , ∑ ((𝑊𝑇 𝑊)
𝑖=1
𝑊
[𝑚
]
𝛿𝑖 /2
𝐴 𝑖) ] [
= 𝐼.
∑ ((𝑊𝑇 𝑊) 𝐴 𝑖 )]
[𝑖=1
]
(8)
𝑇
In this case, 𝑋 = 𝑃𝑇 Γ2 𝑃 is a solution of (1).
Proof. Suppose that the matrix equation (1) has a solution.
Then from Theorem 4, 𝐴 𝑖 , 𝑖 = 1, 2, . . . , 𝑚, have the form
𝑚
𝑚
𝑖=1
𝑖=1
−𝛿𝑖 /2
∑ 𝐴 𝑖 = ∑ (𝑊𝑇 𝑊)
Let
𝑚
𝛿𝑖 /2
∑ ((𝑊𝑇 𝑊)
𝑖=1
𝑚
𝐴 𝑖 ) = ∑ 𝑍𝑖 .
(9)
𝑖=1
Then
𝑚
𝑚
𝑖=1
𝑖=1
∑ 𝐴 𝑖 = ∑ (𝑊𝑇 𝑊)
−𝛿𝑖 /2
𝑍𝑖 ,
(10)
𝑈𝑖+1 Γ𝑃.
𝑍𝑖 .
(14)
Since
𝑊
𝑍1
( 𝑍2 )
..
.
𝑍𝑚
(15)
Journal of Applied Mathematics
3
is column-orthonormal, it can be extended to an orthogonal
matrix
𝑊 𝑈
𝑍1 𝑉1
( 𝑍2 𝑉2 ) ,
..
..
.
.
𝑍𝑚 𝑉𝑚
(16)
0
𝑈1
𝑊 𝑈
0 𝑈2
𝑍1 𝑉1
.. ) ( Γ −Σ) (𝑃 0 ) ,
( 𝑍2 𝑉2 ) = ( ...
.
Σ Γ
0 𝑄
..
..
.
.
𝑈𝑚 0
0 𝑈𝑚+1
𝑍𝑚 𝑉𝑚
(17)
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.
Thus 𝑊 = 𝑈1 Γ𝑃, and
𝑚
𝑚−1
∑ 𝑍𝑖 = ∑ 𝑈𝑖+1 Σ𝑃 + ∑ 𝑈𝑖+1 Γ𝑃.
𝑖=1
𝑖=1,3
(18)
𝑖=2,4
𝑚
𝑚
𝑖=1
𝑖=1
−𝛿𝑖 /2
𝑚
= ∑ (𝑃𝑇 Γ2 𝑃)
𝑖=1,3
𝑚−1
(19)
𝑇
𝑋 + ∑ 𝐴𝑇𝑖 𝑋𝛿𝑖 𝐴 𝑖 = (𝑈1 Γ𝑃) (𝑈1 Γ𝑃)
−𝛿𝑖 /2
+ ∑ ((𝑃𝑇 Γ2 𝑃)
𝑖=1,3
𝑇 2
−𝛿𝑖 /2
𝑖=2,4
−𝛿𝑖 /2
𝑇
𝛿𝑖
𝑈𝑖+1 Σ𝑃) (𝑃𝑇 Γ2 𝑃)
−𝛿2 /2
× ((𝑃𝑇 Γ2 𝑃)
−𝛿2 /2
𝑇
𝛿2
𝑈3 Γ𝑃)
−𝛿𝑚−1 /2
𝛿𝑚−1
𝑇
𝑇 2
𝛿𝑖
𝑈𝑖+1 Γ𝑃) (𝑃 Γ 𝑃)
𝑈𝑚 Γ𝑃)
((𝑃𝑇 Γ2 𝑃)
𝑇
−𝛿𝑚−1 /2
𝑈𝑚 Γ𝑃)
−𝛿1 /2
= 𝑃𝑇 Γ2 𝑃 + (𝑃𝑇 Σ𝑈2𝑇 ) (𝑃𝑇 Γ2 𝑃)
𝛿1
−𝛿1 /2
× (𝑃𝑇 Γ2 𝑃) (𝑃𝑇 Γ2 𝑃)
𝑈2 Σ𝑃
−𝛿3 /2
𝛿𝑚
−𝛿1 /2
𝑈4 Σ𝑃
−𝛿𝑚 /2
× (𝑃𝑇 Γ2 𝑃) (𝑃𝑇 Γ2 𝑃)
𝛿2
−𝛿2 /2
+ (𝑃𝑇 Γ𝑈𝑚𝑇 ) (𝑃𝑇 Γ2 𝑃)
𝛿𝑚−1
−𝛿𝑚 /2
𝑈𝑚+1 Σ𝑃
−𝛿2 /2
× (𝑃𝑇 Γ2 𝑃) (𝑃𝑇 Γ2 𝑃)
𝑈3 Γ𝑃
−𝛿𝑚−1 /2
−𝛿𝑚−1 /2
(𝑃𝑇 Γ2 𝑃)
𝑈𝑚 Γ𝑃
=𝑃𝑇 Γ2 𝑃+𝑃𝑇 Σ𝑈2𝑇 𝑈2 Σ𝑃+𝑃𝑇 Σ𝑈4𝑇 𝑈4 Σ𝑃
𝑇
+ ⋅ ⋅ ⋅ +𝑃𝑇 Σ𝑈𝑚+1
𝑈𝑚+1 Σ𝑃
=𝑃𝑇 Γ2 𝑃 + 𝑃𝑇 Σ2 𝑃 + 𝑃𝑇 Σ2 𝑃
+ ⋅ ⋅ ⋅ + 𝑃𝑇 Σ2 𝑃 + 𝑃𝑇 Γ2 𝑃 + ⋅ ⋅ ⋅ + 𝑃𝑇 Γ2 𝑃
=𝑃𝑇 (
𝑈𝑖+1 Γ𝑃)
= 𝑃𝑇 Γ2 𝑃 + ((𝑃𝑇 Γ2 𝑃)
𝑈𝑚+1 Σ𝑃)
𝑈3 Γ𝑃) (𝑃𝑇 Γ2 𝑃)
+ ⋅ ⋅ ⋅ + ((𝑃𝑇 Γ2 𝑃)
× (𝑃𝑇 Γ2 𝑃)
−𝛿𝑚 /2
𝑇
+𝑃𝑇 Γ𝑈3𝑇 𝑈3 Γ𝑃+ ⋅ ⋅ ⋅ +𝑃𝑇 Γ𝑈𝑚𝑇 𝑈𝑚 Γ𝑃
𝑈𝑖+1 Σ𝑃)
+ ∑ ((𝑃 Γ 𝑃)
× ((𝑃𝑇 Γ2 𝑃)
+ ((𝑃𝑇 Γ2 𝑃)
× (𝑃𝑇 Γ2 𝑃)
𝑖=1
𝑚−1
((𝑃𝑇 Γ2 𝑃)
+ (𝑃𝑇 Γ𝑈3𝑇 ) (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Γ𝑃.
Conversely, suppose that 𝐴 𝑖 , 𝑖 = 1, 2, . . . , 𝑚 have the
decomposition (13). Let 𝑋 = 𝑃𝑇 Γ2 𝑃. Then
−𝛿𝑖 /2
𝛿𝑚
𝑈𝑚+1 Σ𝑃)
𝑇
+ . . . + (𝑃𝑇 Σ𝑈𝑚+1
) (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Σ𝑃
𝑖=2,4
× ((𝑃𝑇 Γ2 𝑃)
+ ⋅ ⋅ ⋅ + ((𝑃𝑇 Γ2 𝑃)
× (𝑃𝑇 Γ2 𝑃)
𝛿3
𝑈4 Σ𝑃)
𝛿3
−𝛿𝑖 /2
𝑚
−𝛿3 /2
𝑇
𝑈4 Σ𝑃) (𝑃𝑇 Γ2 𝑃)
× (𝑃𝑇 Γ2 𝑃) (𝑃𝑇 Γ2 𝑃)
+ ∑ (𝑃𝑇 Γ2 𝑃)
𝑚
× ((𝑃𝑇 Γ2 𝑃)
𝑈2 Σ𝑃)
−𝛿3 /2
𝑍𝑖
−𝛿𝑖 /2
−𝛿3 /2
−𝛿1 /2
+ (𝑃𝑇 Σ𝑈4𝑇 ) (𝑃𝑇 Γ2 𝑃)
Then (4) can be rewritten as follows:
∑ 𝐴 𝑖 = ∑ (𝑊𝑇 𝑊)
+ ((𝑃𝑇 Γ2 𝑃)
−𝛿𝑚 /2
and applying Theorem 3, there exist orthogonal matrices
𝑈1 , 𝑈2 , . . . , 𝑈𝑚+1 , 𝑃, 𝑄 and diagonal matrices Γ > 0 and Σ ≥ 0
such that
𝑚
𝛿1
× (𝑃𝑇 Γ2 𝑃) ((𝑃𝑇 Γ2 𝑃)
𝑇
𝑈2 Σ𝑃)
𝑚+1 2
(Γ + Σ2 )) 𝑃 = 𝑃𝑇 𝑃 = I.
2
(20)
This shows that 𝑋 is a solution to the matrix equation (1).
4
Journal of Applied Mathematics
𝑚
𝑚−1
󵄩 󵄩−𝛿𝑖 /2
󵄩 󵄩−𝛿𝑖 /2
= ∑ 󵄩󵄩󵄩󵄩Γ2 󵄩󵄩󵄩󵄩
‖Σ‖ + ∑ 󵄩󵄩󵄩󵄩Γ2 󵄩󵄩󵄩󵄩
‖Γ‖
4. Main Results
𝑖=1,3
In this section, we explain some properties of (1) and we
obtain relations between the solution 𝑋 and the matrices 𝐴 𝑖 .
𝑚−1
𝑖=1,3
𝑖=2,4
(22)
Since we choose −1 < 𝛿𝑖 < 0, then we have ‖Γ‖−𝛿𝑖 ≤ ‖Γ‖ < 1,
and therefore
𝑚
󵄩󵄩 𝑚 󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩∑ 𝐴 𝑖 󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑖=1 󵄩󵄩󵄩
󵄩󵄩 𝑚
󵄩󵄩
−𝛿𝑖 /2
= 󵄩󵄩󵄩 ∑ (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Σ𝑃
󵄩󵄩𝑖=1,3
󵄩
󵄩󵄩
󵄩󵄩
−𝛿𝑖 /2
𝑈𝑖+1 Γ𝑃󵄩󵄩󵄩
+ ∑ (𝑃𝑇 Γ2 𝑃)
󵄩󵄩
𝑖=2,4
󵄩
󵄩󵄩 𝑚
󵄩󵄩
󵄩󵄩
󵄩󵄩
−𝛿𝑖 /2
≤ 󵄩󵄩󵄩 ∑ (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Σ𝑃󵄩󵄩󵄩
󵄩󵄩𝑖=1,3
󵄩󵄩
󵄩
󵄩
𝑚
≤ ∑ ‖Γ‖−𝛿𝑖 ‖Σ‖ + ∑ ‖Γ‖−𝛿𝑖 ‖Γ‖ .
Theorem 6. If the matrix equation (1) has a positive definite
solution 𝑋, then ‖ ∑𝑚
𝑖=1 𝐴 𝑖 ‖ < 𝑚.
Proof. Suppose that (1) has a solution. Then by Theorem 5,
∑𝑚
𝑖=1 𝐴 𝑖 has the decomposition (13).
Then
𝑖=2,4
∑ ‖Γ‖−𝛿𝑖 ≤ (
𝑖=1,3
𝑚+1
),
2
𝑚−1
∑ ‖Γ‖−𝛿𝑖 ≤ (
𝑖=2,4
𝑚−1
).
2
(23)
Thus,
󵄩󵄩 𝑚 󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩∑ 𝐴 𝑖 󵄩󵄩󵄩 ≤ ( 𝑚 + 1 ) (√ 2 )
󵄩󵄩
󵄩
2
𝑚+1
󵄩󵄩𝑖=1 󵄩󵄩󵄩
𝑚−1
+(
(21)
󵄩󵄩 𝑚−1
󵄩󵄩
󵄩󵄩
󵄩󵄩
−𝛿𝑖 /2
+ 󵄩󵄩󵄩 ∑ (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Γ𝑃󵄩󵄩󵄩
󵄩󵄩𝑖=2,4
󵄩󵄩
󵄩
󵄩
𝑚 󵄩
−𝛿𝑖 /2 󵄩
󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩 󵄩󵄩𝑈𝑖+1 󵄩󵄩󵄩󵄩 ‖Σ‖ ‖𝑃‖
≤ ∑ 󵄩󵄩󵄩󵄩(𝑃𝑇 Γ2 𝑃)
󵄩󵄩
󵄩
𝑖=1,3
𝑚−1 󵄩
−𝛿𝑖 /2 󵄩
󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩 󵄩󵄩𝑈𝑖+1 󵄩󵄩󵄩󵄩 ‖Γ‖ ‖𝑃‖ .
+ ∑ 󵄩󵄩󵄩󵄩(𝑃𝑇 Γ2 𝑃)
󵄩󵄩
󵄩
𝑖=2,4
2
2
𝑚−1
) (√
) = 𝑚√
< 𝑚.
2
𝑚+1
𝑚+1
(24)
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:
−𝛿𝑖
𝑇
> ∑𝑚
(i) ∑𝑚
𝑖=1 𝑋
𝑖=1 𝐴 𝑖 𝐴 𝑖 ,
Since each 𝑈𝑖+1 , 𝑖 = 1, 2, . . . , 𝑚, is orthogonal, also 𝑃 is
orthogonal. So
𝑇 𝛿𝑖 +1
(𝛿𝑖 +1)/2
(ii) 𝐼 − ∑𝑚
𝐴 𝑖 − ∑𝑚
𝐴 𝑖 𝐴𝑇𝑖 𝑋(𝛿𝑖 +1)/2 > 0,
𝑖=1 𝐴 𝑖 𝑋
𝑖=1 𝑋
√
(iii) 𝑟(∑𝑚
𝑖=1 𝐴 𝑖 ) ≤ (1/2) 2/(𝑚 + 1)((3𝑚 − 1)/(𝑚 + 1)),
󵄩󵄩 𝑚 󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩∑ 𝐴 𝑖 󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑖=1 󵄩󵄩󵄩
𝑚 󵄩
−𝛿𝑖 /2 󵄩
󵄩
󵄩󵄩
󵄩󵄩 ‖Σ‖
≤ ∑ 󵄩󵄩󵄩󵄩(𝑃𝑇 Γ2 𝑃)
󵄩󵄩
𝑖=1,3 󵄩
𝑚−1 󵄩
−𝛿𝑖 /2 󵄩
󵄩
󵄩󵄩
󵄩󵄩 ‖Γ‖
+ ∑ 󵄩󵄩󵄩󵄩(𝑃𝑇 Γ2 𝑃)
󵄩󵄩
𝑖=2,4 󵄩
𝑚
󵄩
󵄩−𝛿𝑖 /2
≤ ∑ 󵄩󵄩󵄩󵄩𝑃𝑇 Γ2 𝑃󵄩󵄩󵄩󵄩
‖Σ‖
𝑖=1,3
𝑚−1
󵄩
󵄩−𝛿𝑖 /2
+ ∑ 󵄩󵄩󵄩󵄩𝑃𝑇 Γ2 𝑃󵄩󵄩󵄩󵄩
‖Γ‖
𝑖=2,4
(1+𝛿𝑖 )/2
𝑇 (1+𝛿𝑖 )/2
(iv) 𝑟(∑𝑚
𝐴 𝑖 − ∑𝑚
) ≤ 1,
𝑖=1 𝑋
𝑖=1 𝐴 𝑖 𝑋
(1+𝛿𝑖 )/2
𝑇 (1+𝛿𝑖 )/2
(v) 𝑟(∑𝑚
𝐴 𝑖 + ∑𝑚
) ≤ 1.
𝑖=1 𝑋
𝑖=1 𝐴 𝑖 𝑋
Proof. (i) By Theorem 4, we have 𝑋 = 𝑊𝑇 𝑊 and ∑𝑚
𝑖=1 𝐴 𝑖 =
−𝛿𝑖 /2
𝑇
∑𝑚
𝑖=1 (𝑊 𝑊)
𝑚
𝑚
𝑖=1
𝑖=1
𝑍𝑖 . Then
∑ 𝑋−𝛿𝑖 − ∑ 𝐴 𝑖 𝐴𝑇𝑖
𝑚
−𝛿𝑖
= ∑ (𝑊𝑇 𝑊)
𝑖=1
𝑚
−𝛿𝑖 /2
− ∑ (𝑊𝑇 𝑊)
𝑖=1
𝑍𝑖 𝑍𝑖𝑇 (𝑊𝑇 𝑊)
−𝛿𝑖 /2
Journal of Applied Mathematics
𝑚
= ∑ (𝑊𝑇 𝑊)
−𝛿𝑖 /2
5
−𝛿𝑖 /2
1/2
(𝑊𝑇 𝑊)
= (𝑊𝑇 𝑊)
𝑖=1
𝑚
(𝐼 − ∑ 𝑍𝑖 𝑍𝑖𝑇 )
𝑖=1
𝑚
− ∑ (𝑊𝑇 𝑊)
−𝛿𝑖 /2
𝑖=1
𝑚
= ∑ (𝑊𝑇 𝑊)
−𝛿𝑖 /2
𝑍𝑖 𝑍𝑖𝑇 (𝑊𝑇 𝑊)
× (𝑊𝑇 𝑊)
[𝐼 − ∑ 𝑍𝑖 𝑍𝑖𝑇 ] (𝑊𝑇 𝑊)
𝑖=1
−𝛿𝑖 /2
(26)
> 0,
𝑖=1
(25)
𝑇
= 1, 2, . . . , 𝑚, and 𝐼−∑𝑚
𝑖=1 𝑍𝑖 𝑍𝑖
𝑚
𝑇
− ∑𝑖=1 𝑍𝑖 𝑍𝑖 > 0. Thus, part (i)
since 𝜆(𝑍𝑖 𝑍𝑖𝑇 )
𝑇
𝑚
+ ∑ 𝑍𝑖𝑇 (𝐼 − 𝑊𝑇 𝑊) 𝑍𝑖 > 0.
𝑖=1
𝑚
−𝛿𝑖 /2
1/2
𝜆(𝑍𝑖𝑇 𝑍𝑖 ), 𝑖
=
𝑊 𝑊 > 0, and, hence, 𝐼
proved.
𝑇
(ii) Replacing 𝐼 by 𝑊𝑇 𝑊 + ∑𝑚
𝑖=1 𝑍𝑖 𝑍𝑖 , we get
This completes the proof of (ii).
(iii) Using Theorem 6, we get
𝑚
𝑚
𝑖=1
𝑖=1,3
𝜆 (∑ 𝐴 𝑖 ) = 𝜆 ( ∑ (𝑃𝑇 Γ2 𝑃)
=
is
−𝛿𝑖 /2
𝑚−1
𝑈𝑖+1 Σ𝑃
−𝛿𝑖 /2
+ ∑ (𝑃𝑇 Γ2 𝑃)
𝑖=2,4
𝑈𝑖+1 Γ𝑃) ,
𝑚
𝑚
𝑚
𝑖=1
𝑖=1
𝑟 (∑ 𝐴 𝑖 )
𝑖=1
𝐼 − ∑ 𝐴𝑇𝑖 𝑋𝛿𝑖 +1 𝐴 𝑖 − ∑ 𝑋(𝛿𝑖 +1)/2 𝐴 𝑖 𝐴𝑇𝑖 𝑋(𝛿𝑖 +1)/2
𝑚
𝑚
𝑖=1
𝑖=1
󵄨󵄨
𝑚
󵄨󵄨
−𝛿𝑖 /2
= max 󵄨󵄨󵄨𝜆 ( ∑ (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Σ𝑃
𝑖 󵄨󵄨
󵄨 𝑖=1,3
= 𝑊𝑇 𝑊 + ∑ 𝑍𝑖𝑇 𝑍𝑖 − ∑ 𝐴𝑇𝑖 𝑋𝛿𝑖 +1 𝐴 𝑖
𝑚−1
− ∑ 𝑋(𝛿𝑖 +1)/2 𝐴 𝑖 𝐴𝑇𝑖 𝑋(𝛿𝑖 +1)/2
𝑖=2,4
𝑚
𝑖=1
𝑖=1
−𝛿𝑖 /2
= 𝑊𝑇 𝑊 + ∑ 𝑍𝑖𝑇 𝑍𝑖 − ∑ ((𝑊𝑇 𝑊)
𝛿𝑖 +1
𝑇
× (𝑊 𝑊)
𝑚
− ∑(𝑊𝑇 𝑊)
(𝛿𝑖 +1)/2
−𝛿𝑖 /2
((𝑊𝑇 𝑊)
𝑖=1
× ((𝑊𝑇 𝑊)
−𝛿𝑖 /2
𝑇
𝑍𝑖 ) (𝑊𝑇 𝑊)
𝑍𝑖 )
𝑇
𝑇
−𝛿𝑖 /2
((𝑊 𝑊)
𝑍𝑖 )
𝑍𝑖 )
(𝛿𝑖 +1)/2
𝑚
+ ∑ (𝑃𝑇 Γ2 𝑃)
−𝛿𝑖 /2
𝑚−1 󵄩
−𝛿𝑖 /2 󵄩
󵄩
󵄩
+ ∑ 󵄩󵄩󵄩󵄩(𝑃𝑇 Γ2 𝑃)
Γ󵄩󵄩󵄩󵄩 .
󵄩
𝑖=2,4 󵄩
𝑖=1
𝛿𝑖 +1
× (𝑊𝑇 𝑊)
𝑚
− ∑(𝑊𝑇 𝑊)
(𝛿𝑖 +1)/2
−𝛿𝑖 /2
(𝑊𝑇 𝑊)
−𝛿𝑖 /2
((𝑊𝑇 𝑊)
𝑖=1
× (𝑍𝑖𝑇 (𝑊𝑇 𝑊)
= (𝑊𝑇 𝑊)
1/2
−𝛿𝑖 /2
(𝑊𝑇 𝑊)
) (𝑊𝑇 𝑊)
1/2
𝑍𝑖
(27)
Since we choose −1 < 𝛿𝑖 < 0, then we have 0 < (−𝛿𝑖 )/2 <
𝑍𝑖 )
1/2 < 1, and therefore (𝑃𝑇 Γ2 𝑃)
1, 2, . . . , 𝑚; then
(𝛿𝑖 +1)/2
𝑚
𝑟 (∑ 𝐴 𝑖 ) ≤
𝑚
𝑖=1
+ ∑ 𝑍𝑖𝑇 𝑍𝑖
1/2
− ∑ 𝑍𝑖𝑇 (𝑊𝑇 𝑊) 𝑍𝑖 − (𝑊𝑇 𝑊)
𝑖=1
𝑚
1/2
∑ 𝑍𝑖 𝑍𝑖𝑇 (𝑊𝑇 𝑊)
𝑖=1
=
−𝛿𝑖 /2
≤ (𝑃𝑇 Γ2 𝑃), 𝑖 =
𝑚 + 1 󵄩󵄩 𝑇 2
󵄩
󵄩󵄩(𝑃 Γ 𝑃) Σ󵄩󵄩󵄩
󵄩
2 󵄩
+
𝑖=1
𝑚
−𝛿𝑖 /2
𝑚 󵄩
−𝛿𝑖 /2 󵄩
󵄩
󵄩
= ∑ 󵄩󵄩󵄩󵄩(𝑃𝑇 Γ2 𝑃)
Σ󵄩󵄩󵄩󵄩
󵄩
󵄩
𝑖=1,3
𝑖=1
𝑚
󵄩󵄩
󵄩󵄩
𝑈𝑖+1 Γ𝑃)󵄩󵄩󵄩
󵄩󵄩
𝑖=2,4
󵄩
󵄩󵄩 𝑚
󵄩󵄩
󵄩󵄩
󵄩󵄩
−𝛿𝑖 /2
≤ 󵄩󵄩󵄩 ∑ (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Σ𝑃󵄩󵄩󵄩
󵄩󵄩𝑖=1,3
󵄩󵄩
󵄩
󵄩
󵄩󵄩 𝑚−1
󵄩󵄩
󵄩󵄩
󵄩󵄩
−𝛿𝑖 /2
+ 󵄩󵄩󵄩 ∑ (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Γ𝑃󵄩󵄩󵄩
󵄩󵄩𝑖=2,4
󵄩󵄩
󵄩
󵄩
𝑚−1
= 𝑊𝑇 𝑊 + ∑ 𝑍𝑖𝑇 𝑍𝑖
− ∑ 𝑍𝑖𝑇 (𝑊𝑇 𝑊)
󵄨󵄨
󵄨󵄨
𝑈𝑖+1 Γ𝑃)󵄨󵄨󵄨
󵄨󵄨
󵄨
󵄩󵄩 𝑚
󵄩󵄩
−𝛿𝑖 /2
≤ 󵄩󵄩󵄩( ∑ (𝑃𝑇 Γ2 𝑃)
𝑈𝑖+1 Σ𝑃
󵄩󵄩 𝑖=1,3
󵄩
𝑖=1
𝑚
−𝛿𝑖 /2
+ ∑ (𝑃𝑇 Γ2 𝑃)
𝑚
𝑚 − 1 󵄩󵄩 𝑇 2
󵄩
󵄩󵄩(𝑃 Γ 𝑃) Γ󵄩󵄩󵄩
󵄩
2 󵄩
𝑚 + 1 󵄩󵄩 𝑇 2 󵄩󵄩 𝑚 − 1 󵄩󵄩 𝑇 3 󵄩󵄩
󵄩󵄩𝑃 ΣΓ 𝑃󵄩󵄩 +
󵄩󵄩𝑃 Γ 𝑃󵄩󵄩
󵄩
󵄩
2 󵄩
2 󵄩
6
Journal of Applied Mathematics
=
≤
≤
𝑚 + 1 󵄩󵄩 2 󵄩󵄩 𝑚 − 1 󵄩󵄩 3 󵄩󵄩
󵄩󵄩ΣΓ 󵄩󵄩 +
󵄩󵄩Γ 󵄩󵄩
2 󵄩 󵄩
2 󵄩 󵄩
𝑚
± (∑ (𝑊𝑇 𝑊)
𝑖=1
𝑚+1
𝑚−1 3
‖ΣΓΓ‖ +
‖Γ‖
2
2
𝑚
1/2
𝑍𝑖 + ∑ 𝑍𝑖𝑇 (𝑊𝑇 𝑊)
)
𝑖=1
1/2
= ((𝑊𝑇 𝑊)
𝑚−1 3
𝑚+1
‖ΣΓ‖ ‖Γ‖ +
‖Γ‖ .
2
2
1/2
𝑚
𝑇
± ∑ 𝑍𝑖 ) ((𝑊𝑇 𝑊)
𝑖=1
1/2
𝑚
± ∑ 𝑍𝑖 ) ≥ 0.
𝑖=1
(31)
(28)
Let Σ = diag(𝜎𝑗 ), Γ = diag(𝛾𝑗 ). Then ((𝑚 + 1)/2)(𝜎𝑗2 + 𝛾𝑗2 ) = 1.
Thus
𝑟 (∑ 𝐴 𝑖 )
𝑖=1
3
𝑚−1
2
2
𝑚+1
√
(√
)
‖ΣΓ‖ +
2
𝑚+1
2
𝑚+1
=
2
2
𝑚+1
𝑚−1
√
max {𝜎𝑗 𝛾𝑗 } +
(√
)
2
𝑚+1 𝑗
2
𝑚+1
3
References
3
𝜎𝑗2 + 𝛾𝑗2 𝑚 − 1
2
2
𝑚+1
√
√
≤
max
+
(
)
2
𝑚+1 𝑗
2
2
𝑚+1
2
1
3𝑚 − 1
= √
(
).
2 𝑚+1 𝑚+1
(29)
(iv) Appling Lemma 1, we get
𝑚
𝑚
𝑖=1
𝑖=1
𝑟 (∑ 𝑋(1+𝛿𝑖 )/2 𝐴 𝑖 − ∑ 𝐴𝑇𝑖 𝑋(1+𝛿𝑖 )/2 )
𝑚
= 𝑟 (∑ (𝑊𝑇 𝑊)
(1+𝛿𝑖 )/2
−𝛿𝑖 /2
(𝑊𝑇 𝑊)
𝑖=1
𝑚
− ∑ 𝑍𝑖𝑇 (𝑊𝑇 𝑊)
−𝛿𝑖 /2
𝑍𝑖
(1+𝛿𝑖 )/2
(𝑊𝑇 𝑊)
)
𝑖=1
𝑚
= 𝑟 (∑ (𝑊𝑇 𝑊)
1/2
𝑖=1
𝑚
𝑍𝑖 − ∑ 𝑍𝑖𝑇 (𝑊𝑇 𝑊)
𝑖=1
𝑚
≤ 𝑟 ((𝑊𝑇 𝑊) + ∑ 𝑍𝑖𝑇 𝑍𝑖 ) = 𝑟 (𝐼) = 1.
𝑖=1
(v) Consider
𝑚
𝑚
𝑖=1
𝑖=1
𝐼 ± (∑ 𝑋(1+𝛿𝑖 )/2 𝐴 𝑖 + ∑ 𝐴𝑇𝑖 𝑋(1+𝛿𝑖 )/2 )
𝑚
= 𝑊𝑇 𝑊 + ∑ 𝑍𝑖𝑇 𝑍𝑖
𝑖=1
1/2
)
5. Conclusion
In this paper, both necessary and sufficient conditions for
𝑇 𝛿𝑖
the nonlinear matrix equation 𝑋 + ∑𝑚
𝑖=1 𝐴 𝑖 𝑋 𝐴 𝑖 = 𝐼 to
have a positive definite solution 𝑋 are given, where −1 <
𝛿𝑖 < 0, 𝐴 𝑖 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 𝐴 𝑖 , 𝑖 = 1, 2, . . . , 𝑚, are presented.
ffk
𝑚
≤
(1+𝛿𝑖 )/2
𝑇 (1+𝛿𝑖 )/2
Therefore, 𝑟(∑𝑚
𝐴 𝑖 + ∑𝑚
) ≤ 1.
𝑖=1 𝑋
𝑖=1 𝐴 𝑖 𝑋
(30)
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