Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 961568, 6 pages
http://dx.doi.org/10.1155/2013/961568
Research Article
The Optimization on Ranks and Inertias of a Quadratic
Hermitian Matrix Function and Its Applications
Yirong Yao
Department of Mathematics, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Yirong Yao; yryao@staff.shu.edu.cn
Received 3 December 2012; Accepted 9 January 2013
Academic Editor: Yang Zhang
Copyright © 2013 Yirong Yao. 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.
We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function 𝑄 − 𝑋𝑃𝑋∗ subject to a
consistent system of matrix equations 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷. As applications, we derive necessary and sufficient conditions for the
solvability to the systems of matrix equations and matrix inequalities 𝐴𝑋 = 𝐶, 𝑋𝐵 = 𝐷, and 𝑋𝑃𝑋∗ = (>, <, ≥, ≤)𝑄 in the Löwner
partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.
1. Introduction
Throughout this paper, we denote the complex number field
stand for the sets
by C. The notations C𝑚×𝑛 and C𝑚×𝑚
ℎ
of all 𝑚 × 𝑛 complex matrices and all 𝑚 × 𝑚 complex
Hermitian matrices, respectively. The identity matrix with an
appropriate size is denoted by 𝐼. For a complex matrix 𝐴,
the symbols 𝐴∗ and 𝑟(𝐴) stand for the conjugate transpose
and the rank of 𝐴, respectively. The Moore-Penrose inverse of
𝐴 ∈ C𝑚×𝑛 , denoted by 𝐴† , is defined to be the unique solution
𝑋 to the following four matrix equations
(1) 𝐴𝑋𝐴 = 𝐴,
(3) (𝐴𝑋)∗ = 𝐴𝑋,
(2) 𝑋𝐴𝑋 = 𝑋,
(4) (𝑋𝐴)∗ = 𝑋𝐴.
ℎ (𝑋) = 𝐴𝑋𝐵𝑋∗ 𝐴∗ + 𝐴𝑋𝐶 + 𝐶∗ 𝑋∗ 𝐴∗ + 𝐷,
(3)
where 𝐵 and 𝐷 are Hermitian matrices. Moreover, Tian [22]
investigated the maximal and minimal ranks and inertias of
the quadratic Hermitian matrix function
(1)
𝑓 (𝑋) = 𝑄 − 𝑋𝑃𝑋∗
Furthermore, 𝐿 𝐴 and 𝑅𝐴 stand for the two projectors 𝐿 𝐴 =
𝐼 − 𝐴† 𝐴 and 𝑅𝐴 = 𝐼 − 𝐴𝐴† induced by 𝐴, respectively. It is
∗
. For 𝐴 ∈ C𝑚×𝑚
, its inertia
known that 𝐿 𝐴 = 𝐿∗𝐴 and 𝑅𝐴 = 𝑅𝐴
ℎ
I𝑛 (𝐴) = (𝑖+ (𝐴) , 𝑖− (𝐴) , 𝑖0 (𝐴))
The investigation on maximal and minimal ranks and
inertias of linear and quadratic matrix function is active
in recent years (see, e.g., [1–24]). Tian [21] considered the
maximal and minimal ranks and inertias of the Hermitian
quadratic matrix function
(2)
is the triple consisting of the numbers of the positive, negative, and zero eigenvalues of 𝐴, counted with multiplicities,
respectively. It is easy to see that 𝑖+ (𝐴) + 𝑖− (𝐴) = 𝑟(𝐴). For
two Hermitian matrices 𝐴 and 𝐵 of the same sizes, we say
𝐴 > 𝐵 (𝐴 ≥ 𝐵) in the Löwner partial ordering if 𝐴 − 𝐵 is
positive (nonnegative) definite.
(4)
such that 𝐴𝑋 = 𝐶.
The goal of this paper is to give the maximal and minimal
ranks and inertias of the matrix function (4) subject to the
consistent system of matrix equations
𝐴𝑋 = 𝐶,
𝑝×𝑝
𝑋𝐵 = 𝐷,
(5)
are given complex matrices.
where 𝑄 ∈ C𝑛×𝑛
ℎ , 𝑃 ∈ Cℎ
As applications, we consider the necessary and sufficient
2
Journal of Applied Mathematics
conditions for the solvability to the systems of matrix equations and inequality
Lemma 2 (see [4]). Let 𝐴 ∈ C𝑚×𝑛 , 𝐵 ∈ C𝑚×𝑘 , 𝐶 ∈ C𝑙×𝑛 , 𝐷 ∈
C𝑚×𝑝 , 𝐸 ∈ C𝑞×𝑛 , 𝑄 ∈ C𝑚1 ×𝑘 , and 𝑃 ∈ C𝑙×𝑛1 be given. Then
𝐴𝑋 = 𝐶,
𝑋𝐵 = 𝐷,
𝑋𝑃𝑋∗ = 𝑄,
(1) 𝑟 (𝐴) + 𝑟 (𝑅𝐴 𝐵) = 𝑟 (𝐵) + 𝑟 (𝑅𝐵 𝐴) = 𝑟 [𝐴 𝐵] ,
𝐴𝑋 = 𝐶,
𝑋𝐵 = 𝐷,
𝑋𝑃𝑋∗ > 𝑄,
𝐴𝑋 = 𝐶,
𝑋𝐵 = 𝐷,
𝑋𝑃𝑋∗ < 𝑄,
𝐴
(2) 𝑟 (𝐴) + 𝑟 (𝐶𝐿 𝐴 ) = 𝑟 (𝐶) + 𝑟 (𝐴𝐿 𝐶) = 𝑟 [ ] ,
𝐶
𝐴𝑋 = 𝐶,
𝑋𝐵 = 𝐷,
𝑋𝑃𝑋∗ ≥ 𝑄,
𝐴𝑋 = 𝐶,
𝑋𝐵 = 𝐷,
𝑋𝑃𝑋∗ ≤ 𝑄,
(6)
(3) 𝑟 (𝐵) + 𝑟 (𝐶) + 𝑟 (𝑅𝐵 𝐴𝐿 𝐶) = 𝑟 [
(4) 𝑟 (𝑃) + 𝑟 (𝑄) + 𝑟 [
in the Löwner partial ordering to be feasible, respectively.
(5) 𝑟 [
2. The Optimization on Ranks and Inertias
of (4) Subject to (5)
In this section, we consider the maximal and minimal ranks
and inertias of the quadratic Hermitian matrix function (4)
subject to (5). We begin with the following lemmas.
, 𝐵 ∈ C𝑚×𝑝 , and 𝐶 ∈ C𝑞×𝑚
Lemma 1 (see [3]). Let 𝐴 ∈ C𝑚×𝑚
ℎ
be given and denote
𝑃1 = [
𝐴 𝐵
],
𝐵∗ 0
𝐴 𝐵 𝐶∗
],
𝑃3 = [ ∗
𝐵 0 0
𝐴 𝐶∗
𝑃2 = [
],
𝐶 0
𝐴 𝐵 𝐶∗
𝑃4 = [
].
𝐶 0 0
max 𝑟 [𝐴 − 𝐵𝑌𝐶 − (𝐵𝑌𝐶)∗ ]
𝐴 𝐵 0
𝐴 𝐵𝐿 𝑄
] = 𝑟 [𝐶 0 𝑃] ,
𝑅𝑃 𝐶 0
[ 0 𝑄 0]
𝐴 𝐷 𝐵
𝑅𝐵 𝐴𝐿 𝐶 𝑅𝐵 𝐷
] + 𝑟 (𝐵) + 𝑟 (𝐶) = 𝑟 [ 𝐸 0 0 ] .
0
𝐸𝐿 𝐶
[𝐶 0 0 ]
(10)
, 𝐵 ∈ C𝑚×𝑛 , 𝐶 ∈ C𝑛×𝑛
Lemma 3 (see [23]). Let 𝐴 ∈ C𝑚×𝑚
ℎ
ℎ ,
𝑚×𝑛
𝑝×𝑛
be given, and, 𝑇 ∈ C𝑚×𝑚 be
𝑄 ∈ C , and 𝑃 ∈ C
nonsingular. Then
(1) 𝑖± (𝑇𝐴𝑇∗ ) = 𝑖± (𝐴) ,
(2) 𝑖± [
𝐴 0
] = 𝑖± (𝐴) + 𝑖± (𝐶) ,
0 𝐶
(3) 𝑖± [
0 𝑄
] = 𝑟 (𝑄) ,
𝑄∗ 0
(4) 𝑖± [
𝐴 𝐵 0
𝐴 𝐵𝐿 𝑃
] + 𝑟 (𝑃) = 𝑖± [𝐵∗ 0 𝑃∗ ] .
∗
𝐿 𝑃𝐵
0
[0 𝑃 0]
(7)
Then
𝐴 𝐵
],
𝐶 0
(11)
Lemma 4. Let 𝐴, 𝐶, 𝐵, and 𝐷 be given. Then the following
statements are equivalent.
𝑌∈C𝑝×𝑞
= min {𝑟 [𝐴 𝐵 𝐶∗ ] , 𝑟 (𝑃1 ) , 𝑟 (𝑃2 )} ,
min 𝑟 [𝐴 − 𝐵𝑌𝐶 − (𝐵𝑌𝐶)∗ ]
(1) System (5) is consistent.
𝑌∈C𝑝×𝑞
= 2𝑟 [𝐴 𝐵 𝐶∗ ]
(2) Let
+ max {𝑤+ + 𝑤− , 𝑔+ + 𝑔− , 𝑤+ + 𝑔− , 𝑤− + 𝑔+ } ,
𝑟 [𝐴 𝐶] = 𝑟 (𝐴) ,
𝑖 [𝐴 − 𝐵𝑌𝐶 − (𝐵𝑌𝐶)∗ ] = min {𝑖± (𝑃1 ) , 𝑖± (𝑃2 )} ,
max
𝑝×𝑞 ±
𝑌∈C
𝐷
[ ] = 𝑟 (𝐵) ,
𝐵
𝐴𝐷 = 𝐶𝐵.
(12)
min 𝑖± [𝐴 − 𝐵𝑌𝐶 − (𝐵𝑌𝐶)∗ ]
𝑌∈C𝑝×𝑞
= 𝑟 [𝐴 𝐵 𝐶∗ ] + max {𝑖± (𝑃1 ) − 𝑟 (𝑃3 ) , 𝑖± (𝑃2 ) − 𝑟 (𝑃4 )} ,
(8)
where
𝑤± = 𝑖± (𝑃1 ) − 𝑟 (𝑃3 ) ,
In this case, the general solution can be written as
𝑋 = 𝐴† 𝐶 + 𝐿 𝐴 𝐷𝐵† + 𝐿 𝐴𝑉𝑅𝐵 ,
(13)
where 𝑉 is an arbitrary matrix over C with appropriate size.
𝑔± = 𝑖± (𝑃2 ) − 𝑟 (𝑃4 ) .
(9)
Now we give the fundamental theorem of this paper.
Journal of Applied Mathematics
3
Theorem 5. Let 𝑓(𝑋) be as given in (4) and assume that 𝐴𝑋 =
𝐶 and 𝑋𝐵 = 𝐷 in (5) is consistent. Then
max
∗
𝑟 (𝑄 − 𝑋𝑃𝑋 )
Note that
∗
𝑟 [𝑄 − (𝑋0 + 𝐿 𝐴 𝑉𝑅𝐵 ) 𝑃(𝑋0 + 𝐿 𝐴 𝑉𝑅𝐵 ) ]
𝐴𝑋=𝐶,𝑋𝐵=𝐷
𝑄
(𝑋0 + 𝐿 𝐴 𝑉𝑅𝐵 ) 𝑃
= 𝑟[
] − 𝑟 (𝑃)
∗
𝑃(𝑋0 + 𝐿 𝐴 𝑉𝑅𝐵 )
𝑃
0
𝑃 𝑃
{
= min {𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵)
∗
∗
{
[−𝐷 0 𝐵 ]
= 𝑟 [[
− 𝑟 (𝑃) , 2𝑛 + 𝑟 (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ )
+([
𝑄 0 𝐷
}
−2𝑟 (𝐴) , 𝑟 [ 0 −𝑃 𝐵 ] − 2𝑟 (𝐵)} ,
∗
∗
[𝐷 𝐵 0 ]
}
min
∗
𝐿𝐴
] 𝑉 [0 𝑅𝐵 𝑃]) ] − 𝑟 (𝑃) ,
0
∗
𝑟 (𝑄 − 𝑋𝑃𝑋∗ )
= 𝑖± [
0
𝑃 𝑃
= 2𝑛 + 2𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 2𝑟 (𝐴) − 2𝑟 (𝐵) − 𝑟 (𝑃)
∗
∗
[−𝐷 0 𝐵 ]
𝑄
(𝑋0 + 𝐿 𝐴 𝑉𝑅𝐵 ) 𝑃
] − 𝑖± (𝑃)
∗
𝑃(𝑋0 + 𝐿 𝐴𝑉𝑅𝐵 )
𝑃
𝑄 𝑋𝑃
𝐿
= 𝑖± [[ ∗ 0 ] + [ 𝐴 ] 𝑉 [0 𝑅𝐵 𝑃]
0
𝑃𝑋0 𝑃
+ max {𝑠+ + 𝑠− , 𝑡+ + 𝑡− , 𝑠+ + 𝑡− , 𝑠− + 𝑡+ } ,
+([
𝑖± (𝑄 − 𝑋𝑃𝑋∗ )
∗
𝐿𝐴
] 𝑉 [0 𝑅𝐵 𝑃]) ] − 𝑖± (𝑃) .
0
(20)
𝐴𝑋=𝐶,𝑋𝐵=𝐷
Let
{
= min {𝑛 + 𝑖± (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ )
{
𝑞 (𝑉) = [
𝑄 0 𝐷
}
−𝑟 (𝐴) , 𝑖± [ 0 −𝑃 𝐵 ] − 𝑟 (𝐵)} ,
∗
∗
[𝐷 𝐵 0 ]
}
min
(19)
𝑖± [𝑄 − (𝑋0 + 𝐿 𝐴𝑉𝑅𝐵 ) 𝑃(𝑋0 + 𝐿 𝐴 𝑉𝑅𝐵 ) ]
𝐴𝑋=𝐶,𝑋𝐵=𝐷
max
𝑄 𝑋0 𝑃
𝐿
] + [ 𝐴] 𝑉 [0 𝑅𝐵 𝑃]
0
𝑃𝑋0∗ 𝑃
𝐿
𝑄 𝑋0 𝑃
] + [ 𝐴] 𝑉 [0 𝑅𝐵 𝑃]
0
𝑃𝑋0∗ 𝑃
∗
𝐿
+ ([ 𝐴 ] 𝑉 [0 𝑅𝐵 𝑃]) .
0
(14)
(21)
Applying Lemma 1 to (19) and (20) yields
𝑖± (𝑄 − 𝑋𝑃𝑋∗ )
max 𝑟 [𝑞 (𝑉)] = min {𝑟 (𝑀) , 𝑟 (𝑀1 ) , 𝑟 (𝑀2 )} ,
𝐴𝑋=𝐶,𝑋𝐵=𝐷
𝑉
0
𝑃 𝑃
= 𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵)
∗
∗
[−𝐷 0 𝐵 ]
(15)
min 𝑟 [𝑞 (𝑉)]
𝑉
= 2𝑟 (𝑀) + max {𝑠+ + 𝑠− , 𝑡+ + 𝑡− , 𝑠+ + 𝑡− , 𝑠− + 𝑡+ } , (22)
− 𝑖± (𝑃) + max {𝑠± , 𝑡± } ,
max 𝑖± [𝑞 (𝑉)] = min {𝑖± (𝑀1 ) , 𝑖± (𝑀2 )} ,
𝑉
where
𝐶𝑃 𝐴𝑄𝐴∗
𝑠± = −𝑛+𝑟 (𝐴)−𝑖∓ (𝑃)+𝑖± (𝐴𝑄𝐴 −𝐶𝑃𝐶 )−𝑟 [ ∗
],
𝐵 𝐷∗ 𝐴∗
∗
min 𝑖± [𝑞 (𝑉)] = 𝑟 (𝑀) + max {𝑠± , 𝑡± } ,
𝑉
∗
𝑄 0 𝐷
0 𝑃 𝐵
𝑡± = −𝑛+𝑟 (𝐴)−𝑖∓ (𝑃)+𝑖± [ 0 −𝑃 𝐵 ] − [𝐴𝑄 𝐶𝑃 0 ] .
∗
∗
∗
∗
[𝐷 𝐵 0 ] [ 𝐷 𝐵 0 ]
(16)
Proof. It follows from Lemma 4 that the general solution of
(4) can be expressed as
𝑋 = 𝑋0 + 𝐿 𝐴 𝑉𝑅𝐵 ,
(17)
where 𝑉 is an arbitrary matrix over C and 𝑋0 is a special
solution of (5). Then
∗
𝑄 − 𝑋𝑃𝑋∗ = 𝑄 − (𝑋0 + 𝐿 𝐴𝑉𝑅𝐵 ) 𝑃(𝑋0 + 𝐿 𝐴𝑉𝑅𝐵 ) .
(18)
where
𝑀=[
𝑄 𝑋0 𝑃 𝐿 𝐴 0
],
0 𝑃𝑅𝐵
𝑃𝑋0∗ 𝑃
𝑄 𝑋0 𝑃 0
𝑀2 = [𝑃𝑋0∗ 𝑃 𝑃𝑅𝐵 ] ,
[ 0 𝑅𝐵 𝑃 0 ]
𝑄 𝑋0 𝑃 𝐿 𝐴
0 ],
𝑀1 = [𝑃𝑋0∗ 𝑃
0
0]
𝐿
[ 𝐴
𝑄 𝑋0 𝑃 𝐿 𝐴 0
0 𝑃𝑅𝐵 ] ,
𝑀3 = [𝑃𝑋0∗ 𝑃
0
0
0 ]
[ 𝐿𝐴
𝑄 𝑋0 𝑃 𝐿 𝐴 0
0 𝑃𝑅𝐵 ] ,
𝑀4 = [𝑃𝑋0∗ 𝑃
0 ]
[ 0 𝑅𝐵 𝑃 0
𝑠± = 𝑖± (𝑀1 ) − 𝑟 (𝑀3 ) ,
𝑡± = 𝑖± (𝑀2 ) − 𝑟 (𝑀4 ) .
(23)
4
Journal of Applied Mathematics
Applying Lemmas 2 and 3, elementary matrix operations and
congruence matrix operations, we obtain
(c) 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 have a common solution such
that 𝑄 − 𝑋𝑃𝑋∗ > 0 if and only if
𝑖+ (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 𝑟 (𝐴) ≥ 0,
0
𝑃 𝑃
𝑟 (𝑀) = 𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) ,
∗
∗
[−𝐷 0 𝐵 ]
𝑄 0 𝐷
𝑖+ [ 0 −𝑃 𝐵 ] − 𝑟 (𝐵) ≥ 𝑛.
∗
∗
[𝐷 𝐵 0 ]
𝑟 (𝑀1 ) = 2𝑛 + 𝑟 (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 2𝑟 (𝐴) + 𝑟 (𝑃) ,
𝑖± (𝑀1 ) = 𝑛 + 𝑖± (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 𝑟 (𝐴) + 𝑖± (𝑃) ,
(d) 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 have a common solution such
that 𝑄 − 𝑋𝑃𝑋∗ < 0 if and only if
𝑄 0 𝐷
𝑟 (𝑀2 ) = 𝑟 [ 0 −𝑃 𝐵 ] − 2𝑟 (𝐵) + 𝑟 (𝑃) ,
∗
∗
[𝐷 𝐵 0 ]
𝑖− (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 𝑟 (𝐴) ≥ 0,
𝑄 0 𝐷
𝑖− [ 0 −𝑃 𝐵 ] − 𝑟 (𝐵) ≥ 𝑛.
∗
∗
[𝐷 𝐵 0 ]
𝑄 0 𝐷
𝑖± (𝑀2 ) = 𝑖± [ 0 −𝑃 𝐵 ] − 𝑟 (𝐵) + 𝑖± (𝑃) ,
∗
∗
[𝐷 𝐵 0 ]
𝑟 (𝑀3 ) = 2𝑛 + 𝑟 (𝑃) − 2𝑟 (𝐴) − 𝑟 (𝐵) + 𝑟 [
(27)
∗
𝐶𝑃 𝐴𝑄𝐴
],
𝐵∗ 𝐷∗ 𝐴∗
0 𝑃 𝐵
𝑟 (𝑀4 ) = 𝑛 + 𝑟 (𝑃) + 𝑟 [𝐴𝑄 𝐶𝑃 0 ] − 2𝑟 (𝐵) − 𝑟 (𝐴) .
∗
∗
[ 𝐷 𝐵 0]
(24)
Substituting (24) into (22), we obtain the results.
(28)
(e) All common solutions of 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 satisfy
𝑄 − 𝑋𝑃𝑋∗ ≥ 0 if and only if
𝑛 + 𝑖− (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 𝑟 (𝐴) = 0,
𝑄 0 𝐷
or, 𝑖− [ 0 −𝑃 𝐵 ] − 𝑟 (𝐵) = 0.
∗
∗
[𝐷 𝐵 0 ]
(29)
(f) All common solutions of 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 satisfy
𝑄 − 𝑋𝑃𝑋∗ ≤ 0 if and only if
𝑛 + 𝑖+ (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 𝑟 (𝐴) = 0,
Using immediately Theorem 5, we can easily get the
following.
Theorem 6. Let 𝑓(𝑋) be as given in (4), 𝑠± and let 𝑡± be as
given in Theorem 5 and assume that 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 in
(5) are consistent. Then we have the following.
(a) 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 have a common solution such
that 𝑄 − 𝑋𝑃𝑋∗ ≥ 0 if and only if
0
𝑃 𝑃
𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) − 𝑖− (𝑃) + 𝑠− ≤ 0,
∗
∗
[−𝐷 0 𝐵 ]
0
𝑃 𝑃
𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) − 𝑖− (𝑃) + 𝑡− ≤ 0.
∗
∗
[−𝐷 0 𝐵 ]
(25)
(b) 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 have a common solution such
that 𝑄 − 𝑋𝑃𝑋∗ ≤ 0 if and only if
0
𝑃 𝑃
𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) − 𝑖+ (𝑃) + 𝑠+ ≤ 0,
∗
∗
[−𝐷 0 𝐵 ]
0
𝑃 𝑃
𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) − 𝑖+ (𝑃) + 𝑡+ ≤ 0.
∗
∗
[−𝐷 0 𝐵 ]
(26)
𝑄 0 𝐷
or, 𝑖+ [ 0 −𝑃 𝐵 ] − 𝑟 (𝐵) = 0.
∗
∗
[𝐷 𝐵 0 ]
(30)
(g) All common solutions of 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 satisfy
𝑄 − 𝑋𝑃𝑋∗ > 0 if and only if
0
𝑃 𝑃
𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) − 𝑖+ (𝑃) + 𝑠+ = 𝑛,
∗
∗
[−𝐷 0 𝐵 ]
(31)
or
0
𝑃 𝑃
𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) − 𝑖+ (𝑃) + 𝑡+ = 𝑛.
∗
∗
[−𝐷 0 𝐵 ]
(32)
(h) All common solutions of 𝐴𝑋 = 𝐶 and 𝑋𝐵 = 𝐷 satisfy
𝑄 − 𝑋𝑃𝑋∗ < 0 if and only if
0
𝑃 𝑃
𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) − 𝑖− (𝑃) + 𝑠− = 𝑛,
∗
∗
[−𝐷 0 𝐵 ]
(33)
or
0
𝑃 𝑃
𝑛 + 𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] − 𝑟 (𝐴) − 𝑟 (𝐵) − 𝑖− (𝑃) + 𝑡− = 𝑛.
∗
∗
[−𝐷 0 𝐵 ]
(34)
Journal of Applied Mathematics
5
(i) 𝐴𝑋 = 𝐶, 𝑋𝐵 = 𝐷, and 𝑄 = 𝑋𝑃𝑋∗ have a common
solution if and only if
max
𝑖− (𝑄 − 𝑋𝑋∗ )
𝐴𝑋=𝐶,𝑋𝐵=𝐷
= min {𝑛 + 𝑖− (𝑇2 ) − 𝑟 (𝐴) , 𝑛 + 𝑖− (𝑇3 ) − 𝑟 (𝐵)} ,
0
𝑃 𝑃
2𝑛+2𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] −2𝑟 (𝐴)−2𝑟 (𝐵)−𝑟 (𝑃)+𝑠+ +𝑠− ≤ 0,
∗
∗
[−𝐷 0 𝐵 ]
0
𝑃 𝑃
2𝑛+2𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] −2𝑟 (𝐴)−2𝑟 (𝐵)−𝑟 (𝑃)+𝑡+ +𝑡− ≤ 0,
∗
∗
[−𝐷 0 𝐵 ]
0
𝑃 𝑃
2𝑛+2𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] −2𝑟 (𝐴)−2𝑟 (𝐵)−𝑟 (𝑃) + 𝑠+ +𝑡− ≤ 0,
∗
∗
[−𝐷 0 𝐵 ]
0
𝑃 𝑃
2𝑛+2𝑟 [ 𝐴𝑄 𝐶𝑃 0 ] −2𝑟 (𝐴)−2𝑟 (𝐵)−𝑟 (𝑃) + 𝑠− +𝑡+ ≤ 0.
∗
∗
[−𝐷 0 𝐵 ]
(35)
Let 𝑃 = 𝐼 in Theorem 5, we get the following corollary.
Corollary 7. Let 𝑄 ∈ C𝑛×𝑛 , 𝐴, 𝐵, 𝐶, and 𝐷 be given. Assume
that (5) is consistent. Denote
𝐶 𝐴𝑄
𝑇1 = [ ∗ ∗ ] ,
𝐵 𝐷
𝑇2 = 𝐴𝑄𝐴 − 𝐶𝐶 ,
𝑄 𝐷
𝑇3 = [ ∗ ∗ ] ,
𝐷 𝐵 𝐵
𝐶 𝐴𝑄𝐴∗
𝑇4 = [ ∗ ∗ ∗ ] ,
𝐵 𝐷 𝐴
𝑇5 = [
∗
𝑖+ (𝑄 − 𝑋𝑋∗ )
= 𝑟 (𝑇1 ) + max {𝑖+ (𝑇2 ) − 𝑟 (𝑇4 ) , 𝑖+ (𝑇3 ) − 𝑛 − 𝑟 (𝑇5 )} ,
min
𝑖− (𝑄 − 𝑋𝑋∗ )
𝐴𝑋=𝐶,𝑋𝐵=𝐷
= 𝑟 (𝑇1 ) + max {𝑖− (𝑇2 ) − 𝑟 (𝑇4 ) , 𝑖− (𝑇3 ) − 𝑟 (𝑇5 )} .
(37)
Remark 8. Corollary 7 is one of the results in [24].
Let 𝐵 and 𝐷 vanish in Theorem 5, then we can obtain
the maximal and minimal ranks and inertias of (4) subject
to 𝐴𝑋 = 𝐶.
Corollary 9. Let 𝑓(𝑋) be as given in (4) and assume that
𝐴𝑋 = 𝐶 is consistent. Then
max 𝑟 (𝑄 − 𝑋𝑃𝑋∗ )
𝐴𝑋=𝐶
= min {𝑛 + 𝑟 [𝐴𝑄 𝐶𝑃] − 𝑟 (𝐴) − 𝑟 (𝐵) ,
2𝑛 + 𝑟 (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 2𝑟 (𝐴) , 𝑟 (𝑄) + 𝑟 (𝑃)}
min 𝑟 (𝑄 − 𝑋𝑃𝑋∗ )
∗
𝐶𝐵 𝐴𝑄
].
𝐵∗ 𝐵 𝐷∗
min
𝐴𝑋=𝐶,𝑋𝐵=𝐷
𝐴𝑋=𝐶
= 2𝑛 + 2𝑟 [𝐴𝑄 𝐶𝑃] − 2𝑟 (𝐴)
(36)
+ max {𝑠+ + 𝑠− , 𝑡+ + 𝑡− , 𝑠+ + 𝑡− , 𝑠− + 𝑡+ } ,
max 𝑖± (𝑄 − 𝑋𝑃𝑋∗ )
𝐴𝑋=𝐶
= min {𝑛 + 𝑖± (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 𝑟 (𝐴) , 𝑖± (𝑄) + 𝑖∓ (𝑃)} ,
min 𝑖± (𝑄 − 𝑋𝑃𝑋∗ )
Then,
𝐴𝑋=𝐶
max
= 𝑛 + 𝑟 [𝐴𝑄 𝐶𝑃] − 𝑟 (𝐴) + 𝑖∓ (𝑃) + max {𝑠± , 𝑡± } ,
∗
𝑟 (𝑄 − 𝑋𝑋 )
(38)
𝐴𝑋=𝐶,𝑋𝐵=𝐷
= min {𝑛 + 𝑟 (𝑇1 ) − 𝑟 (𝐴) − 𝑟 (𝐵) , 2𝑛 + 𝑟 (𝑇2 )
−2𝑟 (𝐴) , 𝑛 + 𝑟 (𝑇3 ) − 2𝑟 (𝐵)} ,
min
∗
𝑟 (𝑄 − 𝑋𝑋 )
𝐴𝑋=𝐶,𝑋𝐵=𝐷
= 2𝑟 (𝑇1 ) + max {𝑟 (𝑇2 ) − 2𝑟 (𝑇4 ) , −𝑛 + 𝑟 (𝑇3 )
− 2𝑟 (𝑇5 ) , 𝑖+ (𝑇2 ) + 𝑖− (𝑇3 )
− 𝑟 (𝑇4 ) − 𝑟 (𝑇5 ) , −𝑛 + 𝑖− (𝑇2 )
+𝑖+ (𝑇3 ) − 𝑟 (𝑇4 ) − 𝑟 (𝑇5 )}
max
𝑖+ (𝑄 − 𝑋𝑋∗ )
𝐴𝑋=𝐶,𝑋𝐵=𝐷
= min {𝑛 + 𝑖+ (𝑇2 ) − 𝑟 (𝐴) , 𝑖+ (𝑇3 ) − 𝑟 (𝐵)} ,
where
𝑠± = − 𝑛 + 𝑟 (𝐴) − 𝑖∓ (𝑃)
+ 𝑖± (𝐴𝑄𝐴∗ − 𝐶𝑃𝐶∗ ) − 𝑟 [𝐶𝑃 𝐴𝑄𝐴∗ ] ,
(39)
𝑡± = −𝑛 + 𝑟 (𝐴) + 𝑖± (𝑄) − 𝑟 (𝑃) − [𝐴𝑄 𝐶𝑃] .
Remark 10. Corollary 9 is one of the results in [22].
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