R E S E A R C H
Open Access
The maximal and minimal ranks of matrix
expression with applications
Zhiping Xiong, Yingying Qin
*and Shifang Yuan
* Correspondence: qinyy04@163. com
Department of Mathematics, Wuyi University, Jiangmen 529020, P.R. China
Abstract
We give in this article the maximal and minimal ranks of the matrix expression A-B1V1C1-B2V2C2-B3V3C3-B4V4C4with respect toV1,V2, V3, andV4. As applications, we
derive the extremal ranks of the generalized Schur complementA - BM(1)C - DN(1)G
and the partial matrix (A BM(1)C DN(1)G) with respect to the generalized inverseM(1) ε M{1} andN(1)ÎN{1}.
AMS classifications:15A03; 15A09; 15A24.
Keywords:matrix expression, maximal rank, minimal rank, generalized inverse, rank equality
1 Introduction
LetCm × nbe the set of allm × ncomplex matrices with complex entries.Indenotes the identity matrix of ordernandOm × ndenotes them×nmatrix of all zero entries (if no confusion occurs, we will omit the subscript). For a given a matrix AÎ Cm× n,
the symbolsA* and r(A) will stand for the conjugate transpose and the rank of the
matrixA, respectively. We recall that a generalized inverse X ÎCn× mofAÎCm × n is a matrix which satisfies some of the following four Penrose equations [1]:
(1)AXA=A, (2)XAX=X, (3) (AX)∗=AX, (4) (XA)∗ =XA.
For a subset {i,j,...,k} of the set {1,2,3,4}, the set ofn × mmatrices satisfying the equa-tions (i), (j), ...,(k) from among the above four Penrose Equations (1)-(4) is denoted by A{i,j,...,k}. A matrixX fromA{i,j,...,k} is called an {i,j,...,k}-inverse ofAand is denoted by A(i,j,...,k)
. In particular, an n × mmatrix X of the setA{1} is called a g-inverse ofAand denoted byA(1). The unique {1, 2, 3, 4}-inverse of Ais denoted byA†, which is called
the Moore-Penrose inverse ofA. Throughout this article, the abbreviated symbolsEA
andFAstand for the two projectorsEA= I - AA†andFA = I−A†AofA, respectively. We refer the reader to [2,3] for basic results on the generalized inverses.
Given a matrix with some variant entries in it (often called partial matrix) or a matrix expression with some variant matrices in it, the rank of the partial matrix or matrix expression will vary with respect to the variant entries or variant matrices. Because the rank of matrix is an integer between 0 and the minimal of row and col-umn numbers of the matrix, maximal and minimal ranks of partial matrix or matrix expressions must exist with respect to their variant entries or variant matrices. Many problems in matrix theory and applications are closely related to maximal and minimal
possible ranks of matrix expressions with variant entries. For example, a matrix equa-tion AXB = Cis consistent if and only if the minimal rank ofC-AXBwith respect toX is zero, see [4-6]; there is matrixX such that the partial matrixAXBof ordernis non-singular if and only if the maximal rank ofAXBwith respect toX isn, see [7-11].
The maximal and minimal ranks of matrix expressions or partial matrix are two basic concepts in matrix theory for describing the dimension of the row or column vector space of matrix expressions or partial matrix, both of which are well understood and are easy to compute by the well-known elementary or congruent matrix opera-tions, see [5,7,8,10-16]. These two quantities play an essential role in characterizing algebraic properties of matrices expressions or partial matrices. In fact, maximal and minimal ranks of matrix expressions or partial matrices have been the main objects of study in matrix theory and applications. Some previous systematical researches on maximal and minimal ranks of matrix expressions or partial matrices and their applica-tions can be found in [17-20]. In recent years, the present author reconsidered the maximal and minimal ranks of matrix expressions or partial matrices by using some tricky operations on block matrices and generalized inverses of matrices, and obtained many new formulas for maximal and minimal ranks of matrix expressions or partial matrices and their applications, see [4,6,9,21-28].
In this article, given matrices A∈Cm×n, B
i∈Cm×pi, Ci∈Cqi×n, i= 1, 2, 3, 4, we will
present the maximal and minimal ranks of the matrix expressionA-B1V1C1- B2V2C2
-B3V3C3 - B4V4C4with respect to V1,V2,V3, andV4. As applications, the maximal and minimal ranks of the generalized Schur complementA - BM(1)C - DN(1)Gand the par-tial matrix (A BM(1)C DN(1)G) with respect to the generalized inverseM(1)Î M{1} and N(1)ÎN
{1} are also considered. The results in this article extend the earlier studies by various authors, see, e.g., [4-6,11,16,18,21,25,26].
We first introduce some well-known results which will be used in this article. Lemma 1.1[5,8,25]. Let
M= ⎛
⎝AA1121AA1222AX23
Y A32A33
⎞ ⎠
where Aij∈Cmi×nj(1≤i, j≤3) are given, X∈Cm1×n3 and Y∈Cm3×n1 are two var-iant matrices. Then
max
X,Y r(M) = min
m3+n3+r
A11A12
A21A22
, m1+n1+r
A22A23
A32A33
,
m1+m3+r(A21A22A23), n1+n3+r
⎛ ⎝AA1222
A32
⎞ ⎠ ⎫ ⎬ ⎭,
(1)
minr
X,Y (M) =r(A21 A22A23) +r ⎛ ⎝AA1222
A32
⎞ ⎠+ max
r
A11A12
A21A22
−r
A12
A22
−r(A21A22),
r
A22A23
A32A33
−r
A22
A32
−r(A22A23)
.
(2)
A(1)=A†+ (In−A†A)W+Z(Im−AA†), (3)
whereWÎCn × mandZÎ Cn×mare arbitrary.
Lemma 1.3[9]. Let AÎ Cm × n,BÎ Cm × k, andCÎCl × n. Then
(1).r(A B) =r(A) +r(EAB) =r(EBA) +r(B),
(2).r
A C
=r(A) +r(CFA) =r(AFC) +r(C),
whereEA =Im-AA†andFA=In-A†A.
2 The maximal and minimal ranks of A - B1V1C1 -B2V2C2 - B3V3C3 - B4V4C4 In this section, we will present the maximal and minimal ranks of the linear matrix expression
P(V1,V2,V3,V4) =A−B1V1C1−B2V2C2−B3V3C3−B4V4C4, (4)
where A∈Cm×n, B
i∈Cm×pi, Ci∈Cqi×n, i= 1, 2, 3, 4, are given matrices, with
respect to four variant matrices Vi∈Cpi×qi, i= 1, 2, 3, 4. Applying the formula (1) in
Lemma 1.1 to the linear matrix expression in (4) and simplifying, we obtain the follow-ing result.
Theorem 2.1Let P(V1,V2,V3,V4) be given as (4). Then
max
V1,V2,V3,V4
r(P(V1, V2, V3, V4)) = min{T1,T2,T3,T4}, (5)
where
r(P(V1,V2,V3,V4)) =r
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
O O O O O O O IP4−V4
O O O O C4 O O O Iq4
O O O O O IP2−V2 O O
O O O O C2 O Iq2 O O
O B3 O B1 A B2 O B4 O
O O Iq1 O C1 O O O O
O O−V1IP1 O O O O O
Iq3 O O O C3 O O O O
−V3IP3 O O O O O O O
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
−
4
i=1
pi−
4
i=1
qi,
=r(T)−
4
i=1
pi−
4
i=1
qi,
Proof. It is easy to verify by block Gaussian elimination that the rank ofP(V1,V2,V3, V4) in (4) can be expressed as
r(P(V1,V2,V3,V4)) =r
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
O O O O O O O IP4−V4
O O O O C4 O O O Iq4
O O O O O IP2−V2 O O
O O O O C2 O Iq2 O O
O B3 O B1 A B2 O B4 O
O O Iq1 O C1 O O O O
O O −V1IP1 O O O O O
Iq3 O O O C3 O O O O
−V3IP3 O O O O O O O
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
−
4
i=1
pi−
4
i=1
qi,
=r(T)−
4
i=1
pi−
4
i=1
where A∈Cm×n, B
i∈Cm×pi, Ci∈Cqi×n, Vi∈Cpi×qi, i= 1, 2, 3, 4 and Ipi,Iqi,i= 1, 2, 3, 4, are denotes the identity matrix of orderpiand qi, respectively.
T= ⎛
⎝ EO E1 S E2−V34
−V3E4 O
⎞ ⎠,S=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
O O O C4 O O O
O O O O Ip2−V2 O O O O C2 O Iq2 O B3 O B1 A B2 O B4
O Iq1 O C1 O O O O −V1Ip1 O O O O
O O O C3 O O O
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and
E1= (O O O O O O Iq3)∗, E2= (O O O O O O Ip4), E3= (Iq4O O O O O O)∗, E4= (Ip3O O O O O O). According to this result, we have
max
V1,V2,V3,V4
r(P(V1,V2,V3,V4)) = max
V1,V2,V3,V4 r(T)−
4
i=1
pi−
4
i=1
qi. (6)
Then applying the formula (1) in Lemma 1.1 to matrix T, we have
max
V3,V4
r(T) = min
p3+q4+r
O E2
E1 S
p4+q3+r
S E3
E4 O
,
p4+p3+r(E1S E2), q3+q4+r
⎛ ⎝ES2
E4
⎞ ⎠ ⎫ ⎬ ⎭
= minp3+q4+p4+q3+r(S1), p4+q3+q4+p3+r(S2),
p4+p3+q4+q3+r(S3), q3+q4+p4+p3+r(S4)
,
where
S1=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
O O O O IP2−V2 O O O C4 O O
O O O C2 O Iq2 O B3B1 A B2 O
Iq1 O O C1 O O
−V1 O Ip1 O O O
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, S2=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
O O O IP2 O −V2 O O C2 O O Iq2 O B1 A B2B4 O
Iq1 O C1 O O O O O C3 O O O
−V1Ip1 O O O O ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
S3
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
O O O O IP2 O −V2 O O O C2 O O Iq2 O B3B1 A B2B4 O
Iq1 O O C1 O O O
−V1 O Ip1 O O O O ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, S4=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
O O O IP2−V2 O O C4 O O
O O C2 O Iq2 O B1 A B2 O
Iq1 O C1 O O O O C3 O O
−V1Ip1 O O O
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Again applying the formula (1) in Lemma 1.1, we get
max
V1,V2,V3,V4
r(T) = min
p3+q4+p4+q3+ max
V1,V2
r(S1),p4+q3+q4+p3+ max
V1,V2
r(S2) ,
p4+p3+q4+q3+ max
V1,V2
r(S3),q3+q4+p4+p3+ max
V1,V2
r(S4)
and
max V1,V2
r(S1) = min
⎧ ⎨
⎩p1+q2+q1+p2+r
⎛ ⎝O O CO O C42
B3B1 A
⎞
⎠,p1+q2+q1+p2+r
⎛ ⎝BO C3 A B4 O2
O C1 O
⎞ ⎠,
p1+q2+q1+p2+r
O O C4O
B3B1 A B2
,p1+q2+q1+p2+r
⎛ ⎜ ⎜ ⎝
O C4
O C2
B3 A
O C1
⎞ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭,
(8)
max V1,V2
r(S2) = min
⎧ ⎨
⎩p1+q2+q1+p2+r
⎛ ⎝BO C1 A B2O4
O C3O
⎞
⎠,p1+q2+q1+p2+r
⎛ ⎝CA B1O O2B4
C3O O
⎞ ⎠,
p1+q2+q1+p2+r
B1 A B2B4
O C3O O
,p1+q2+q1+p2+r
⎛ ⎜ ⎜ ⎝
C2O
A B4
C1O
C3O
⎞ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭,
(9)
max
V1,V2
r(S3) = min
p1+q2+q1+p2+r
O O C2 O
B3B1 A B4
,
p1+q2+q1+p2+r
B3 A B2B4
O C1 O O
,
p1+q2+q1+p2+r(B3B1A B2B4),
p1+q2+q1+p2+r
⎛
⎝BO C3 A B2 O4
O C1 O
⎞ ⎠,
(10)
max
V1,V2
r(S4) = min
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
p1+q2+q1+p2+r
⎛ ⎜ ⎜ ⎝
O C4
O C2
B1 A
O C3
⎞ ⎟ ⎟
⎠,p1+q2+q1+p2+r
⎛ ⎜ ⎜ ⎝
C4 O
A B2
C1 O
C3 O
⎞ ⎟ ⎟ ⎠,
p1+q2+q1+p2+r
⎛ ⎝BO C1 A B4 O2
O C3 O
⎞
⎠,p1+q2+q1+p2+r
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
C4
C2
A C1
C3
⎞ ⎟ ⎟ ⎟ ⎟ ⎠.
(11)
Substituting (8)-(11) into (7) and (6) yield (5).
Recall a simple fact that a matrix equation AXB = Cis consistent for every variant
matrices X, if and only if the maximal rank of C - AXBwith respect to X is zero.
Thus, by Theorem 2.1 we can immediately obtain the following result.
Corollary 2.2 LetP(V1,V2,V3, V4) be given as (4). Then the matrix equation A = B1V1C1 + B2V2C2 + B3V3C3 + B4V4C4 holds for anyV1, V2,V3, andV4 if and only if T1= OorT2=OorT3 =OorT4=O.
Because the right side of (5) are just composed by ranks of block matrices, they can be easily simplified by block Gaussian elimination when the given matrices in (4) satisfy some restrictions.
Theorem 2.3 LetP(V1,V2,V3,V4) be given as (4) and letR(B1)⊆R(B2),R(B3)⊆R (B4), R(B1)⊆R(B2),R(B3)⊆R(B4),R(C∗2)⊆R(C∗1),R(C∗4)⊆R(C∗3) Then
max
V1,V2,V3,V4
where
τ1= min
⎧ ⎨ ⎩r
⎛
⎝O O CO O C42
B3B1 A
⎞ ⎠, r
O C4 O
B3 A B2
, r
⎛ ⎝BO C3 A4
O C1
⎞ ⎠ ⎫ ⎬ ⎭,
τ2= min
r
O C2 O
B1 A B4
, r(A B2B4), r
A B4
C1 O
,
τ3= min
⎧ ⎨ ⎩r
⎛ ⎝BO C1 A2
O C3
⎞ ⎠, r
A B2
C3 O
, r
⎛ ⎝CA1
C3 ⎞ ⎠ ⎫ ⎬ ⎭.
Proof. In fact, we can write B1=B2X,B3=B4Y,C2=ZC1, andC4= WC3under the hypotheses of Theorem 2.3. In this case, we have
r
O O C4 O
B3B1 A B2
=r
O C4 O
B3 A B2
, r ⎛ ⎜ ⎜ ⎝
O C4
O C2
B3 A
O C1
⎞ ⎟ ⎟ ⎠=r
⎛ ⎝BO C3 A4
O C1
⎞ ⎠,
r ⎛
⎝O O OO O C2
B3B1 A
⎞ ⎠=r
⎛
⎝O OO O ZCC41
B3B2X A
⎞ ⎠≤r
⎛
⎝BO C3 A B4 O2
O C1 O
⎞ ⎠ (13) and r
B1 A B2B4
O C3 O O
=r
A B2B4
C3 O O
, r ⎛ ⎜ ⎜ ⎝
C2 O
A B4
C1 O
C3 O
⎞ ⎟ ⎟ ⎠=r
⎛ ⎝CA B1 O4
C3 O
⎞ ⎠,
r ⎛
⎝BO C1 A B2 O4
O C3 O
⎞ ⎠=r
⎛
⎝BO ZC2X A B1 O4
O C3 O
⎞ ⎠≤r
⎛
⎝CA B1 O O2B4
C3 O O
⎞ ⎠
(14)
and
r(B3B1A B2B4) =r(A B2B4) =r
⎛
⎝BO C3 A B2 O4
O C1 O
⎞ ⎠=r
A B4
C1 O
,
r
O O C2 O
B3B1 A B4
=r
O O ZC1 O
B3B2X A B4
≤r
B3 A B2B4
O C1 O O
,
r
O O C2 O
B3B1 A B4
=r
O C2 O
B1 A B4
(15)
and
r
⎛
⎝BO C1 A B4 O2
O C3 O
⎞ ⎠=r
A B2
C3 O
,r ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ C4 C2 A C1 C3 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠=r
⎛ ⎝CA1
C3
⎞ ⎠,r
⎛ ⎜ ⎜ ⎝
O C4
O C2
B1 A
O C3
⎞ ⎟ ⎟ ⎠=r
⎛ ⎝BO C1 A2
O C3
⎞ ⎠, r ⎛ ⎜ ⎜ ⎝
O C4
O C2
B1 A
O C3
⎞ ⎟ ⎟ ⎠=r
⎛ ⎜ ⎜ ⎝
O C4
O ZC1
B2X A
O C3
⎞ ⎟ ⎟ ⎠≤r
⎛ ⎜ ⎜ ⎝
C4 O
A B2
C1 O
C3 O
⎞ ⎟ ⎟ ⎠
Combining (5) with (13)-(16) yields (12).
Corollary 2.4 Let P(V1, V2, V3, V4) be given as (4) and let
R(B1)⊆R(B2),R(B3)⊆R(B4),R(C∗2)⊆R(C∗1),R(C∗4)⊆R(C∗3) then the matrix
equa-tion A = B1V1C1+ B2V2C2+ B3V3C3 + B4V4C4 holds for anyV1,V2,V3, andV4 if and only ifτ1= Oorτ2 =Oorτ3=O.
In the rest of this section, we will find the minimal rank of the linear matrix
expres-sion P(V1,V2, V3, V4) in (4), with respect to four variant matrices
Vi∈Cpi×qi, i= 1, 2, 3, 4, whenP(V1,V2,V3,V4) satisfy some restrictions.
Theorem 2.5 Let P(V1, V2, V3, V4) be given as (4) and let
R(B1)⊆R(B2),R(B3)⊆R(B4),R(C∗2)⊆R(C∗1),R(C∗4)⊆R(C∗3). Then
min
V1,V2,V3,V4
r(P(V1,V2,V3,V4))
=r(A B2B4) +r
A B4
C1 O
+r
O C2 O
B1 A B4
+r
A B2
C3 O
+r
⎛ ⎝BO C1 A2
O C3
⎞ ⎠
+r ⎛ ⎝CA1
C3
⎞ ⎠−r
B1 A B4
O C1 O
−r
O C2 O
B2 A B4
−r ⎛ ⎝BO C1 A1
O C3
⎞ ⎠−r
⎛ ⎝CA B2 O2
C3 O
⎞ ⎠
+max ⎧ ⎨ ⎩r
O C4 O
B3 A B2
+r
⎛ ⎝BO C3 A4
O C1
⎞ ⎠+r
⎛
⎝O O CO O C42
B3B1 A
⎞ ⎠−r
⎛
⎝O O CO O C41
B3B1 A
⎞ ⎠
−r ⎛
⎝O O CO O C42
B3B2 A
⎞
⎠−β1−β2, r
A B2B4
C3 O O
+r
⎛ ⎝CA B1 O4
C3 O
⎞ ⎠
+r ⎛
⎝BO C1 A B2 O4
O C3 O
⎞ ⎠−r
⎛
⎝BO C1 A B1 O4
O C3 O
⎞ ⎠−r
⎛
⎝CA B2 O O2B4
C3 O O
⎞ ⎠−2β3
⎫ ⎬ ⎭,
(17)
where
β1= min
⎧ ⎨ ⎩r
⎛
⎝BO O C3B1 A2
O O C3
⎞ ⎠, r
B3 A B2
O C3 O
, r
⎛ ⎝BO C3 A1
O C3
⎞ ⎠ ⎫ ⎬ ⎭,
β2= min
⎧ ⎨ ⎩r
⎛
⎝O CO C42 OO
B1 A B4
⎞ ⎠, r
C4 O O
A B2B4
, r
⎛ ⎝CA B4 O4
C1 O
⎞ ⎠ ⎫ ⎬ ⎭,
β3= min
⎧ ⎨ ⎩r
⎛
⎝BO C1 A B2 O4
O C3 O
⎞ ⎠, r
A B2B4
C3 O O
, r
⎛ ⎝CA B1 O4
C3 O
⎞ ⎠ ⎫ ⎬ ⎭.
Proof. From the proof of Theorem 2.1, it is easy to verify that the minimal rank ofP (V1, V2,V3,V4) in (4) can be expressed as
min
V1,V2,V3,V4
r(P(V1,V2,V3,V4)) = min
V1,V2,V3,V4 r(T)−
4
i=1
pi−
4
i=1
qi, (18)
where T, S, Ei, piand qi, i = 1,2,3,4, are given as the proof of Theorem 2.1. Then
min
V1,V2,V3,V4
r(T) = min
V3,V4 r
⎛
⎝ EO E1 S E2−V34
−V3E4 O
⎞
⎠=r(E1 S E3) +r
⎛ ⎝ES2
E4
⎞ ⎠
+ max
r
O E2
E1 S
−r
E2
S
−r(E1 S),
r
S E3
E4 O
−r
S E4
−r(S E3)
.
(19)
In this case, we derive from (19) that
min
V1,V2,V3,V4
r(T) = min
V1,V2
r(E1 S E3) + min
V1,V2 r
⎛ ⎝ES2
E4
⎞ ⎠
+ max
min
V1,V2 r
O E2
E1 S
−min
V1,V2 r
E2
S
−min
V1,V2
r(E1 S),
min
V1,V2 r
S E3
E4 O
−min
V1,V2 r
S E4
−min
V1,V2
r(S E3)
.
(20)
Again applying the formula (2) in Lemma 1.1, we have
min
V1,V2
r(E1 S E3) =q4+q3+ min
V1,V2 r(S3)
=
4
i=1
qi+p1+p2+r(B3 B1 A B2 B4) +r
⎛
⎝BO C3 A B2 O4
O C1 O
⎞ ⎠
+ max ⎧ ⎨ ⎩r
O O C2 O
B3B1 A B4
−r ⎛
⎝BO O C3B1 A B2 O4
O O C1 O
⎞ ⎠−r
O O C2 O O
B3B1 A B2B4
,
r
B3 A B2B4
O C1 O O
−r
⎛
⎝BO C3 A B2 O O2B4
O C1 O O
⎞ ⎠−r
B3B1 A B2B4
O O C1 O O
⎫⎬ ⎭,
(21)
where S3is given as the Equation (7) of the proof of Theorem 2.1. Since B1 =B2X,
B3 =B4Y,C2=ZC1, and C4 =WC3, (21) is reduced to
min
V1,V2
r(E1S E3)
=
4
i=1
qi+p1+p2+r(A B2 B4) +r
A B4
C1 O
+ max
r
O C2 O
B1 A B4
−r
B1 A B4
O C1 O
−r
C2 O O
A B2B4
, −r
A B2B4
C1 O O
=
4
i=1
qi+p1+p2+r(A B2 B4) +r
A B4
C1 O
+r
O C2 O
B1 A B4
−r
B1 A B4
O C1 O
−r
C2 O O
A B2B4
.
(22)
r
O C2 O
B1 A B4
=r
O ZC1 O
B2X A B4
=r ⎛ ⎝Z O
O I
O C1 O
B2 A B4
⎛ ⎝O I OX O O
O O I ⎞ ⎠ ⎞ ⎠
≥r
Z O O I
O C1 O
B2 A B4
+r
⎛
⎝O C1 O
B2 A B4
⎛ ⎝O I OX O O
O O I ⎞ ⎠ ⎞ ⎠
−r
O C1 O
B2 A B4
=r
O C2 O
B2 A B4
+r
O C1 O
B1 A B4
−r
O C1 O
B2 A B4
.
With the similar method, we also have
min
V1,V2 r
⎛ ⎝ES2
E4
⎞ ⎠=
4
i=1
pi+q1+q2+r
⎛ ⎝CA1
C3
⎞ ⎠+r
A B2
C3 O
+r
⎛ ⎝BO C1 A2
O C3
⎞ ⎠
−r ⎛ ⎝BO C1 A1
O C3
⎞ ⎠−r
⎛ ⎝CA B2 O2
C3 O
⎞ ⎠,
(23)
min
V1,V2 r
O E2
E1 S
=p1+p2+p4+q1+q2+q3+r
O C4 O
B3 A B2
+r
⎛ ⎝BO C3 A4
O C1
⎞ ⎠
+r ⎛
⎝O O CO O C42
B3B1 A
⎞ ⎠−r
⎛
⎝O O CO O C41
B3B1 A
⎞ ⎠−r
⎛
⎝O O CO O C42
B3B2 A
⎞ ⎠,
(24)
min
V1,V2
r
S E3
E4 O
=q1+q2+q4+p1+p2+p3+r
A B2B4
C3 O O
+r
⎛ ⎝CA B1 O4
C3 O
⎞ ⎠
+r
⎛ ⎝O CB1A B2O4
O C3O
⎞ ⎠−r
⎛ ⎝BO C1 A B1 O4
O C3 O
⎞ ⎠−r
⎛ ⎝CA B2 O O2B4
C3 O O
⎞ ⎠.
(25)
On the other hand, by the formula (1) in Lemma 1.1, we have
min
V1,V2 r
E2
S
=p4+p1+p2+q1+q2+ min
⎧ ⎨ ⎩r
⎛
⎝BO O C3B1 A2
O O C3
⎞ ⎠, r
⎛
⎝BO C3 A B4 O2
O C3 O
⎞ ⎠,
r ⎛ ⎝BO C3 A1
O C3
⎞ ⎠ ⎫ ⎬ ⎭.
max
V1,V2 r
S E4
=p3+p1+p2+q1+q2+ min
⎧ ⎨ ⎩r
⎛
⎝BO C1 A B2 O4
O C3 O
⎞ ⎠,r
⎛
⎝CA B4 O O2B4
C3 O O
⎞ ⎠,
r
⎛ ⎝CA B1 O4
C3 O
⎞ ⎠ ⎫ ⎬ ⎭,
(27)
max
V1,V2
r(E1 S) =q3+p1+p2+q1+q2+ min
⎧ ⎨ ⎩r
⎛
⎝O CO C42 OO
B1 A B4
⎞ ⎠,r
C4 O O
A B2B4
,
r
⎛ ⎝CA B4 O4
C1 O
⎞ ⎠ ⎫ ⎬ ⎭,
(28)
max
V1,V2
r(S E3) =q3+p1+p2+q1+q2+ min
⎧ ⎨ ⎩r
⎛
⎝BO C1 A B2 O4
O C3 O
⎞ ⎠,r
A B2B4
C3 O O
,
r
⎛ ⎝CA B1 O4
C3 O
⎞ ⎠ ⎫ ⎬ ⎭.
(29)
Contrasting (18), (20) and (22)-(29) yields (17).
Corollary 2.6 LetP(V1, V2, V3,V4) be given as (4) and letR(B1)⊆R(B2), R(B3)⊆R (B4), R(B1)⊆R(B2),R(B3)⊆R(B4),R(C∗2)⊆R(C∗1),R(C∗4)⊆R(C∗3). Then the matrix
equation A = B1V1C1 +B2V2C2+ B3V3C3 +B4V4C4 is consistent if and only if the right
side of (17) is zero.
3 Some applications to generalized Schur complement and partial matrix
As direct applications of the results in Section 2, we determine in this section the
max-imal and minmax-imal ranks of the generalized Schur complement A-BM(1)C-DN(1)Gand
the partial matrix (A BM(1)C DN(1)G) with respect to two variant matricesM(1)ÎM{1} and N(1)ÎN{1}.
Theorem 3.1Let AÎ Cm × n,BÎCm × p,CÎCq × n, DÎCm × s, GÎCt × n,MÎ Cq × p, andNÎCt × s.
Then
max
M(1)∈M{1},N(1)∈N{1}r(A−BM
(1)
C−DN(1)G) = min{T1, T2, T3}, (30)
where
T1= min
⎧ ⎨ ⎩r
⎛ ⎝A D BG N O
C O M
⎞
⎠−r(M)−r(N),r
A D B G N O
−r(N), r
⎛ ⎝A DG N
C O
⎞ ⎠−r(N)
⎫ ⎬ ⎭,
T2= min
A B D C M O
−r(M),r(A B D),r
A D C O
,
T3= min
⎧ ⎨ ⎩r
⎛ ⎝A BC M
G O
⎞
⎠−r(M), r
⎛ ⎝AC
G
⎞ ⎠, r
A B G O
Proof. Applying Lemma 1.2, we have
M(1)=M†+FMW1+W2EM (31)
and
N(1)=N†+FNW3+W4EN, (32)
where Wi,i = 1,2,3,4 are arbitrary,EM=Iq - MM†andFM=Ip- M†M. Substituting
the Equation (31) and Equation (32) into the generalized Schur complement A - BM(1)
C - DN(1)G yields
A−BM(1)C−DN(1)G=A1−BFMW1C−BW2EMC−DFNW3G−DW4ENG, (33)
whereA1=A - BM†C - DN†G.
In fact A1 - BFMW1C - BW2EMC - DFNW3G - DW4ENGis a special case of the
matrix expressionP(V1,V2,V3,V4), andR(BFM)⊆R(B),R(DFN)⊆ R(D),R((EMC)*) ⊆R (C*),R((ENG)*)⊆R(G*). In this case, from the formula (12) in Theorem 2.3, we have
max
M(1)∈M{1},N(1)∈N{1}r(A−BM
(1)C−DN(1)G)
= max
W1,W2,W3,W4
r(A1−BFMW1C−BW2EMC−DFNW3G−DW4ENG)
= min{T1,T2,T3},
(34)
where
T1 = min ⎧ ⎨ ⎩r
⎛
⎝ OO O EO ENMGC DFNBFM A1
⎞ ⎠r
O ENG O DFN A1 B
,r
⎛
⎝DFO EN AN1G
O C
⎞ ⎠ ⎫ ⎬ ⎭
T2 = min
r
O EMC O BFMA1 D
, r(A1B B), r
A1D
C O
T3 = min ⎧ ⎨ ⎩r
⎛
⎝BFO EM AM1C
O G
⎞ ⎠, r
A1B
G O
, r ⎛ ⎝AC1
G ⎞ ⎠ ⎫ ⎬ ⎭
ForT1′, simplifying the ranks of matrices by Lemma 1.3 and block Gaussian elimina-tion, we find that:
r ⎛
⎝ OO O EO EMNGC DFNBFM A1
⎞ ⎠=
⎛
⎝EAM1G ODFNBFOM EMC O O
⎞ ⎠
=r ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
A1D B O O O
O N O O O O O O M O O O G O O O N O C O O O O M
⎞ ⎟ ⎟ ⎟ ⎟
⎠−2r(N)−2r(M)
=r ⎛ ⎝G N OA D B
C O M ⎞
⎠−r(N)−r(M),
r
O ENG O DFN A1 B
=r
A1 DFN B ENG O O
=r
⎛
⎝AO N O O1D B O G O O N
⎞
⎠−2r(N)
=r
A D B G N O
−r(N),
(36)
r ⎛
⎝DFO EN AN1G
O C
⎞ ⎠=
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
O G N O O C O M D A1O O
N O O O O C O O
⎞ ⎟ ⎟ ⎟ ⎟
⎠−2r(N) =r ⎛ ⎝A DG N
C O ⎞
⎠−r(N). (37)
Combining the rank equalities (35), (36) with (37), we have T1=T1.
By the similar approach, we also have T2 =T2 and T3 =T3. Then we have complete
the proof of theorem.
Corollary 3.2LetAÎCm × n,BÎCm × p,CÎCq × n,DÎCm × s,GÎ Ct × n,MÎ Cq × p, andNÎ Ct × s. Then the identityA = BM(1)C + DN(1)G
holds for everyM(1)Î M{1} and N(1)ÎN{1} if and only if T1=O or T2=O or T3=O.
From the proof of Theorem 3.1, we known that A - BM(1)C - DN(1)G = A1
-BFmW1C - BW2EmC - DFNW3G - DW4ENG, where A1 = A - BM†C - DN†G. In this case, A - BM(1)C - DN(1)Gis a special case of the matrix expressionP(V1,V2,V3,V4), and R(BFM)⊆R(B),R(DFN)⊆R(D),R((EMC)*)⊆R(C*),R((ENG)*)⊆R(G*). Then from the Theorem 2.5, we have
Theorem 3.3Let AÎ Cm × n,BÎCm × p,CÎCq × n, DÎCm × s, GÎCt × n,MÎ Cq × p, andNÎCt × s.
Then
min
M(1)∈M{1},N(1)∈N{1}r(A−BM
(1)C−DN(1)G)
= min
W1,W2,W3,W4
r(A1−BFMW1C−BW2EMC−DFNW3G−DW4ENG)
=r(A B D) +r
A D C O
+r
M C O B A D
+r
⎛ ⎝AC
G ⎞ ⎠+r
A B G O
+r
⎛ ⎝M CB A
O G ⎞ ⎠
−r ⎛ ⎝M O OB A D
O C O ⎞ ⎠−r
O C O M B A D O
−r
⎛ ⎜ ⎜ ⎝
B A M O O C O G
⎞ ⎟ ⎟ ⎠−r
⎛ ⎝C O MA B O
G O O ⎞ ⎠+ 3r(M)
+ max ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r
N G O D A B
+r
⎛ ⎝O G NB A O
O C O ⎞ ⎠+r
⎛ ⎝N O GO M C
D B A ⎞ ⎠−r
⎛ ⎜ ⎜ ⎝
N O G O O C D B A O M O
⎞ ⎟ ⎟ ⎠
−r ⎛ ⎝N O G OO O C M
D B A O ⎞
⎠+r(N)−δ1−δ2,
r
A B D G O O
+r
⎛ ⎝A BC O
G O ⎞ ⎠+r
⎛ ⎝M C OB A D
O G O ⎞ ⎠−r
⎛ ⎜ ⎜ ⎝
B A D M O O O C O O G O ⎞ ⎟ ⎟ ⎠
−r ⎛ ⎝C O O MA B D O
G O O O ⎞ ⎠−2δ3
⎫ ⎬ ⎭,
where
δ1= min
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r
⎛ ⎜ ⎜ ⎝
O M C D B A N O O O O G
⎞ ⎟ ⎟
⎠−r(M), r ⎛ ⎝D A BN O O
O G O ⎞ ⎠, r
⎛ ⎜ ⎜ ⎝
D A N O O C O G
⎞ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ,
δ2= min
⎧ ⎨ ⎩r
⎛
⎝M C O OO G O N B A D O
⎞
⎠−r(M), r
G O O N A B D O
, r
⎛ ⎝G O NA D O
C O O ⎞ ⎠ ⎫ ⎬ ⎭,
δ3= min
⎧ ⎨ ⎩r
⎛ ⎝M C OB A D
O G O ⎞
⎠−r(M), r
A B D G O O
, r ⎛ ⎝A DC O
G O ⎞ ⎠ ⎫ ⎬ ⎭.
Corollary 3.4LetAÎCm × n,BÎCm × p,CÎCq × n,DÎCm × s,GÎ Ct × n,MÎ Cq × p, and NÎCt × s. then the identityA=BM(1)C
+DN(1)Gis consistent if and only if the right side of (38) is zero.
Next, we will determine the maximal and minimal ranks of the partial matrix
(A BM(1)C DN(1)G)
with respect toM(1)Î M{1} andN(1)Î N{1}, by applying the results in Section 2. It is quite obvious that the partial matrix (A BM(1)C DN(1)G) may be written as
(A BM(1)C DN(1)G) = (A O O) +BM(1)(O C O) +DN(1)(O O G). (39)
Then from (39) and Theorems 2.3 and 3.1, we have
Theorem 3.5 LetAÎ Cm × n,BÎCm × p,CÎ Cq × n,DÎCm × s, GÎ Ct × n,MÎ Cq × p, andNÎCt × s.
Then
max
M(1)∈M{1},N(1)∈N{1}r(A BM
(1)C DN(1)G) = minT˜
1, T˜2, T˜3
, (40)
where
˜
T1= min
⎧ ⎨ ⎩r
⎛
⎝A O O D BO O G N O O C O O M
⎞
⎠−r(M)−r(N), r
A O D B O G N O
−r(N),
r
A O D O G N
+r(C)−r(N)
,
˜
T2= min
r
A O B D O C M O
−r(M), r(A B D), r(A D) +r(C)
,
˜
T3= min
r
A O B O C M
+r(G)−r(M), r(A B) +r(G), r(A) +r(C) +r(G)
.
Corollary 3.6Let AÎ Cm × n,BÎCm × p,CÎ Cq × n,DÎCm × s,GÎCt × n,MÎ Cq × p, andNÎ Ct × s. then the inclusionR(BM(1)C + DN(1)G
)⊆R(A) holds for every M(1)Î M
{1} andN(1)Î N{1} if and only if T˜1=O orT˜2=O or T˜3=O.
On the other hand, from (39) and Theorems 2.5 and 3.3, we can easily obtain the minimal rank of the partial matrix (A BM(1)C DN(1)G) with respect toM(1)Î M{1} and N(1)ÎN
Theorem 3.7 LetAÎCm × n, BÎ Cm × p,CÎ Cq × n,DÎ Cm × s,GÎ Ct × n,MÎ Cq × p, andNÎCt × s.
Then
min
M(1)∈M{1},N(1)∈N{1}r(A BM
(1)C DN(1)G)
=r(A) +r(A D) +r(A B) +r(A B D) +r
M O C O B A O D
−r
B A M O
−r
B A D M O O
−r
O O C O M B A O D O
+ 3r(M)
+ max
r
N O G O D A O B
+r
O O G N B A O O
+r
⎛
⎝N O O O GO M O C O D B A O O
⎞ ⎠
−r ⎛
⎝N O O GD B A O O M O O
⎞ ⎠−r
⎛
⎝N O O O G OaO O O C O M D B A O O O
⎞
⎠+r(N)−ξ1−ξ2,
r(G) +r(A D) +r(A B D) +r ⎛
⎝M O C OB A O D O G O
⎞ ⎠−r
B A D M O O
−r
O C O O M A O B D O
−2ξ3
,
(41)
where
ξ1= min
⎧ ⎨ ⎩r
⎛
⎝O M O CD B A O N O O O
⎞
⎠+r(G)−r(M), r
D A B N O O
+r(G),
r
D A N O
+r(C) +r(G)
ξ2= min{r
⎛
⎝O O O G O NM O C O O O B A O O D O
⎞
⎠−r(M), r
O G O O N A O B D O
,
r
⎛
⎝O O G O NA O O D O O C O O O
⎞ ⎠,
ξ3= min{r
M O C O B A O D
+r(G)−r(M), r(A B D) +r(G), r(A D) +r(C) +r(G)}.
Corollary 3.8 LetAÎ Cm × n,BÎ Cm × p,CÎ Cq × n,DÎ Cm × s,GÎ Ct × n,M Î Cq × p, andNÎCt × s. then there are someM(1)Î M
{1} andN(1)ÎN{1}, such that the inclusion R(BM(1)C+DN(1)G)⊆R(A) holds if and only if the right side of (41) is zero.
Acknowledgements
The authors would likes to thank the Editor-in-Chief and the anonymous referees for their very detailed comments, which greatly improved the presentation of this article. The research of the Z. Xiong was supported by the start-up fund of Wuyi University Jiangmen 529020, Guangdong province, P.R. China and the foundation for High-Level Talents in Guangdong province, P.R. China. The research of the S.Yuan was supported by Guangdong Natural Science Fund of China (No.10452902001005845).
Authors’contributions
The authors jointly worked on deriving the results. All authors read and approved the final manuscript. Competing interests
Received: 26 August 2011 Accepted: 6 March 2012 Published: 6 March 2012
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